solving multi step equations homework 3

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Solving Multi-Step Linear Equations

Add/Subtract Times/Divide Multi-Step Parentheses Zero/No/All Sol'n

In the previous two pages, we've looked at solving one-step linear equations; that is, equations that require one addition or subtraction, or that require one multiplication or division. However, most linear equations require more than one step in order to find their solution. What steps then should be used, and in what order?

For multi-step linear equations, we'll be using the same steps as we have previously; the only difference is that we won't be done after one step. We'll still have to do at least one more step. In what order should these steps be done? Well, that's going to vary with the equation, but there are some general guidelines which can prove helpful.

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Solving Multi-Step Equations

Solve 7 x + 2 = −54

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The variable is on the left-hand side (LHS) of the equation. It is currently multiplied by seven, and then it has a two added to it. I need to undo the "times seven" and the "plus two".

There is no rule about which "undo" operation I should do first. However, if I first divide through by 7 , I'm definitely going to create fractions. Personally, I prefer to avoid fractions if possible, so I almost always do any plus / minus before any times / divide. I might end up having to deal with fractions anyway, but at least I can put them off until closer to the end of my work.

Starting with the "plus two" first, I'll subtract two from either side of the equation. Only then will I divide through by the seven. My work looks like this:

7x + 2 = -54 -2 -2 ------------ 7x = -56 -- --- 7 7 x = -8

By doing the plus / minus first, I avoided fractions. As you can see, the answer doesn't involve fractions, so I did myself a favor by doing the dividing-through last. My solution is:

x = −8

Formatting your homework and showing your work in the manner I have done above is, in my experience, fairly universally acceptable. However (warning!), it is also a good idea to clearly rewrite your final answer at the end of each exercise, as shown (in purple) above. Don't expect your grader to take the time to dig through your work and try to figure out what you probably meant your answer to be. Format your work so as to make your meaning clear.

Solve −5 x − 7 = 108

In this equation, the variable (on the left-hand side) is multiplied by a minus five, and then a seven is subtracted from it. In hopes (as always!) of avoiding fractions, I'll add seven to either side of the equation first. Only then will I divide through by the minus five. My work looks like this:

-5x - 7 = 108 +7 +7 ------------- -5x = 115 --- --- -5 -5 x = -23

I've shown my work neatly. Now I'll clearly rewrite my solution at the end of my work:

x = −23

Solve 3 x − 9 = 33

The variable (on the left-hand side of the equation) is multiplied by a three, and then a nine is subtracted from it. I'll take care of the nine first, and then the three:

3x - 9 = 33 +9 +9 ----------- 3x = 42 -- -- 3 3 x = 14

In this case, again, my solution has no fractions:

Solve 5 x + 7 x = 72

In this equation, I have two terms on the left-hand side that contain variables. So my first step is to combine these " like terms " on the left. Then I can solve:

5 x + 7 x = 12 x

So now my equation is:

Even though it might initially have looked more complicated, this is actually a one-step equation. I'll solve by dividing through by twelve:

12x = 72 --- -- 12 12 x = 6

My answer is:

Solve 4 x − 6 = 6 x

In this equation, I've got terms with variables on either side of the equation. To solve, I need to get those variable terms all on one side of the equation.

There is no rule saying which of the two terms I should move, the 4 x or the 6 x . However, I've learned from experience that, to avoid negative coefficients on my variables, I should move the x term with the smaller coefficient. That means, in this case, that I'll subtract the 4 x from the left-hand side over to the right-hand side:

4x - 6 = 6x -4x -4x ------------- -6 = 2x

And now I have a one-step equation, which I'll solve by dividing through by two:

-6 = 2x -- -- 2 2 -3 = x

My solution is:

x = −3

In the above exercise, the variable (in my working) ended up on the right-hand side of the equation. This is perfectly okay. The variable is not "required" to end up on the left-hand side of the equation; we're just used to seeing it there. So the result " −3 =  x " is perfectly okay, and means exactly the same thing as " x  = −3 ".

However (warning!), I have heard that some instructors insist that the variable be placed on the left-hand side of the equation in the final answer . (No, I'm not making this up.) So, even though the " −3 =  x " is perfectly valid in the working, those instructors will count this as "wrong" if you leave the answer this way. If you have any doubts about your instructor's formatting preferences, ask now.

Solve 8 x − 1 = 23 − 4 x

In this equation, I've got variables on either side of the equation, and also loose numbers on either side. I need to get the variable terms on one side, and the loose numbers on the other side. Because I'd like to avoid negative coefficients on my variables, I'll be moving the smaller of the two terms; namely, the −4 x that's currently on the right-hand side. To get the loose numbers on the side opposite the variable terms, I'll be moving the −1 that's currently on the left-hand side. There is no particular "right" order for doing these steps; since they're both a matter of adding, people usually do them together in one step. I'll do the variable terms first, and then the loose numbers:

8x - 1 = 23 - 4x +4x +4x ----------------- 12x - 1 = 23 +1 +1 ------------ 12x = 24

At this point, I've got a one-step equation which requires one division to solve:

12x = 24 --- -- 12 12 x = 2

Then my answer is:

If, in the above, I'd done the first two steps at one go, it would have looked like this:

8x - 1 = 23 - 4x +4x +1 +1 +4x ----------------- 12x = 24 --- -- 12 12 x = 2

It's probably a good idea, when you're just starting out, to do each step separately. But once you get comfortable with the process (and are reliably arriving at the correct values), feel free to start combining some steps.

Solve 5 + 4 x − 7 = 4 x − 2 −  x

This equation is all kinds of messy! Before I can solve, I'll need to combine the like terms on either side of the equation:

   5 + 4 x − 7 = 4 x − 2 − x

(5 − 7) + 4 x = (4 x − 1 x ) − 2

−2 + 4 x = 3 x − 2

Now that I've simplified each side of the equation, I can do the solving.

-2 + 4x = 3x - 2 -3x -3x ----------------- -2 + 1x = -2 +2 +2 ----------------- 1x = 0

I added the (usually unstated) 1 to the variable term on the right-hand side of the original equation in order to help me keep track of what I was doing; it isn't "necessary". And it isn't expected on the final answer, which is properly stated as:

It is perfectly fine for x to have a value of zero. Zero is a valid solution. Do not say that this equation has "no solution"; it does indeed have a solution, that solution being x = 0 .

Solve 0.2 x + 0.9 = 0.3 − 0.1 x

This equation solves just like all the other linear equations I've done. It just looks worse because of the decimals. But that's easy to fix!

Whatever is the largest number of decimal places in any of the coefficients, I can multiply through on both sides by " 1 " followed by that number of zeroes. In this case, all of the decimals have one decimal place, so I'll multiply through by 10 :

10(0.2 x + 0.9) = 10(0.3 − 0.1 x )

10(0.2 x ) + 10(0.9) = 10(0.3) − 10(0.1 x )

2 x + 9 = 3 − 1 x

Now I can solve as usual:

2x + 9 = 3 - 1x +1x +1x ----------------- 3x + 9 = 3 -9 -9 ------------ 3x = -6 -- -- 3 3 x = -2

Just because the original equation had decimal places, doesn't mean that I'm stuck working with them. File this trick away for later; it'll come in handy.

x = −2

By the way, if the coefficent with the most decimal places had had two decimal places, then I'd have multiplied through on both sides of the equation by 100 ; for three decimal places, I'd have multiplied through by 1000 ; and so forth.

Solve katex.render("\\boldsymbol{\\color{green}{ \\frac{1}{4}x + 1 = \\frac{1}{6}x + \\frac{1}{2} }}", typed01);

Ick! Fractions! But, just as with the decimals in the previous exercise, I don't have to be stuck with fractions. In this case, I'll be multiplying through to "clear" the denominators, giving me a nicer equation to solve.

This is now a much nicer equation to work with. I'll continue my solution by subtracting the smaller 2 x from either side:

3x + 12 = 2x + 6 -2x -2x ------------------ 1x + 12 = 6 -12 -12 ------------------ 1x = -6

I'll remove the 1 from the variable when I write my final answer:

x = −6

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Multi-Step Equations

How to solve multi-step equations.

The word “multi” means more than two, or many. That’s why solving multi-step equations are more involved than one-step and two-step equations because they require more steps.

The main goal in solving multi-step equations, just like in one-step and two-step equations, is to isolate the unknown variable on one side of the equation while keeping the constant or number on the opposite side.

However, there is no rule on where to keep the variable. It all depends on your preference. The “standard” or usual way is to have it on the left side. But there are cases when it makes more sense to keep it on the right side of the equation.

Since we are dealing with equations, we need to keep in mind that whatever operation we perform on one side must also be applied on the other to keep the equation “balanced”.

This concept of performing the same operation on both sides applies to the four arithmetic operations, namely: addition, subtraction, multiplication, and division. For example, if we add 5 on the left side of the equation we must also add 5 on the right side.

Key steps to remember:

• Get rid of any grouping symbols such as square brackets, parentheses, etc, by applying the Distributive Property of Multiplication over Addition.
• Simplify both sides of the equation, if possible, by combining like terms.
• Decide where you want to keep the variable because that will help you decide where to place the constant.
• Eliminate numbers or variables by applying opposite operations: addition and subtraction are opposite operations as in the case of multiplication and division.

Examples of How to Solve Multi-Step Equations

Example 1: Solve the multi-step equation below.

This is a typical problem in multi-step equations where there are variables on both sides. Notice that there is no parenthesis in this equation and no like terms to combine on both sides of the equation.

Clearly, our first step is to decide where to keep or isolate the unknown variable [latex]x[/latex]. Since [latex]7x[/latex] is “larger” than [latex]2x[/latex], then we might as well keep it to the left side. This means we will have to get rid of the [latex]2x[/latex] on the right side. To do that, we need to subtract both sides of the equation by [latex]2x [/latex] because the opposite of [latex]+2x[/latex] is [latex]-2x[/latex].

After doing so, it is nice to see just the variable x on the left side. This implies that we have to move all the constants to the right side by eliminating [latex]+3[/latex] on the left side. The opposite of [latex]+3[/latex] is [latex]-3[/latex], therefore, we will subtract both sides by [latex]3[/latex].

The last step is to isolate the variable [latex]x[/latex] by itself on the left side of the equation. Since [latex]+5[/latex] is multiplying [latex]x[/latex], then its opposite operation is to divide by [latex]+5[/latex]. So, we are going to divide both sides by [latex]5[/latex], and then we are done!

Example 2: Solve the multi-step equation below.

Our very first step should be to get rid of the parenthesis by applying the Distributive Property of Multiplication over Addition. That is, multiply [latex] -2[/latex] inside each term in the parenthesis [latex](5-4x)[/latex].

Now, it is time to decide where to keep the unknown variable [latex]x[/latex]. If you decide to keep the variable on the left side, that is perfectly fine.

However, for practice, let’s try keeping it on the right side. We should arrive at the same answers.

To get rid of the [latex]-3x[/latex] on the left side, we add both sides by [latex]3x[/latex] since the opposite of [latex]-3x[/latex] is [latex]+3x[/latex]. After we simplify by adding both sides by [latex]3x[/latex], we obtain this less messy equation.

It’s nice to see the variable x just on the right side. So, we have to move all the constants to the left side.

Clearly, the [latex]-10[/latex] on the right must be removed. The opposite of [latex]-10[/latex] is [latex]+10[/latex], therefore, we will add both sides by [latex]10[/latex]. The last step is to isolate the variable [latex]x[/latex] by itself on the right side of the equation.

Since [latex]+11[/latex] is multiplying [latex]x[/latex], then its opposite operation is to divide by [latex]+11[/latex]. So, we are going to divide both sides by [latex]11[/latex] and we are done!

Example 3: Solve the multi-step equation below.

Our first step should be to eliminate the parentheses on BOTH sides of the equation by applying the Distributive property. For the left side, multiply [latex]-4[/latex] inside each term of the parenthesis [latex](4x-8)[/latex] and for the right side, multiply [latex]+3[/latex] inside the parenthesis [latex](-8x-1)[/latex].

Now, before we even decide which side of the equation to isolate the variable, it looks like we have to perform some house cleaning. We need to combine like terms (the x’s ) on the left side of the equation.

Again it doesn’t matter which side to isolate the variable being solved. Say, we decided to keep it on the left.

That means we have to get rid of the [latex]-24x[/latex] on the right side. The opposite of [latex]-24x[/latex] is [latex]+24x[/latex] so we are going to add both sides by [latex]24x[/latex].

Next, we have to move all the constants to the right side of the equation. That [latex]+32[/latex] on the left side must go! The opposite of [latex]+32[/latex] is [latex]-32[/latex], so then we will subtract both sides by [latex]32[/latex].

The last step is to isolate the variable [latex]x[/latex] by itself on the left side of the equation. Since [latex]+5[/latex] is multiplying [latex]x[/latex], then its opposite operation is to divide by [latex]+5[/latex]. And so, let’s divide both sides by [latex]5[/latex] and we are done!

Example 4: Solve the equation [latex]13x – 9x + 20 = 30 + 2[/latex].

Step-by-Step Solution:

1) Combine the variables on the left side of the equation. That is, [latex]13x – 9x=4x[/latex]. Also, simplify the constants on the right side which gives us [latex]30+2=32[/latex].

2) Get rid of [latex]20[/latex] on the left side by subtracting [latex]20[/latex] on both sides of the equation.

3) To solve for [latex]x[/latex], divide both sides by [latex]4[/latex] to get [latex]x=3[/latex].

Example 5: Solve the equation below.

1) Combine like terms on both sides.

2) Subtract [latex]6y[/latex] on both sides to keep the variable [latex]y[/latex] to the left side only.

3) Add [latex]11[/latex] to both sides of the equation.

4) Finally, divide both sides by [latex]-10[/latex] to get the solution.

Example 6: Solve the equation below.

1) Combine the similar terms with variable [latex]m[/latex], and constants on both sides of the equation.

2) Add [latex]5m[/latex] to both sides of the equation. It will keep the variable on the left side and eliminate the variable on the right.

3) Add [latex]14[/latex] to both sides.

4) The last step is to divide the one-step equation by [latex]-3[/latex] to get the value of [latex]m[/latex].

Example 7: Solve the equation [latex]2\left( {x – 5} \right) = 5x + 23[/latex].

1) Eliminate the parenthesis on the left side of the equation by distributing the number outside the parenthesis into the binomial inside.

2) This time, for convenience, we will keep the variable to the right side. To do that, we subtract [latex]2x[/latex] on both sides of the equation.

3) Next, subtract [latex]23[/latex] to both sides of the equation.

4) This leaves to simply divide both sides by the coefficient of [latex]3x[/latex] which is [latex]3[/latex] to get the value of [latex]x[/latex].

Example 8: Solve the equation below.

1) Remove the two parentheses on both sides of the linear equation by applying the Distributive Property of Multiplication over Addition.

2) Combine the constants on both sides. This will clean up the equation tremendously.

3) Add [latex]7h[/latex] to both sides to keep the term with a variable on the left while eliminating the one on the right.

4) Add [latex]57[/latex] on both sides of the equation to keep the constant on the right.

5) Divide both sides by [latex]22[/latex] to get the final solution. That’s it!

You might also like these tutorials:

• Solving One-Step Equations
• One-Step Equations Practice Problems with Answers
• Solving Two-Step Equations

Chapter 2, Lesson 3: Solving Multi-Step Equations

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Multi-Step Equation Worksheets

A huge collection of printable multi-step equations worksheets involving integers, fractions and decimals as coefficients are given here for abundant practice. Solving and verifying equations, applications in geometry and MCQs are included in this section for 7th grade and 8th grade students. We offer some free worksheets too!

Solving Equations Involving Integers: Level 1

In these 'Level 1' worksheets, solve each multi-step equation to find the value of the unknown variable. Five exclusive pdfs are here for practice.

• Download the set

Solving Equations Involving Integers: Level 2

These 'Level 2' multi-step equations may involve a few more steps to arrive the solution than level 1. A plenty of practice worksheets are available here.

Solving Equations Involving Fractions

These printable worksheets have equations whose coefficients are fractions and integers. Solve each multi-step equation. Eight questions are given per worksheet.

Solving Equations Involving Decimals

In these pdf worksheets for grade 7 and grade 8, perform the basic arithmetic operation and solve the multi-step equations having decimal numbers as coefficients.

Solving Equations: Mixed Review

A combination of integer, fraction and decimal coefficients stands for the variable in these mixed review worksheets. Practice them all.

Equation Word Problems Worksheets

Read the word problems and write down the equations. Solve them in one, two or more steps to find the solution.

(30 Worksheets)

Solve and Verify the Solution

In these pdf worksheets, solve the multi-step equations and verify your solution by substituting the value of the unknown variable to the equation.

Translating Multi-Step Equation

Translate the given phrases to algebraic equations. Three exclusive practice sheets are available here for students.

Equations in Geometry: Type 1

Nine geometric shapes are shown in each worksheet. Their sides are given in the form of expressions. Solve them to find the unknown variable.

Area and Perimeter - Shapes: Type 2

In these printable worksheets, the area and perimeter of nine shapes are given. Use the given expressions and apply the area and perimeter formula to solve them.

Equations in Geometry: Type 3

A collection of word problems involving properties of shapes are given. Set up the equation and solve each multi-step equation.

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» Simplifying Algebraic Expressions

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Solving Multi-Step Equations

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Multi Step Equations

Multi step equations are equations that need more than two steps to solve for the variable. We use the same operation on both sides of the equation to such equations. Solving multiple step equations is sometimes complicated when compared to one step or two-step equations as they require multiple steps.

Let us see how to solve multi-step equations and the properties that we use for the same.

What are Multi Step Equations?

Multi step equations are equations that require more than one operation (applied on both sides) to solve for the required variable. Often word problems also may lead to multi step equations. They look complicated . Here are some examples of multi step equations:

• 2 (4x - 5) = 3x - 7
• –3 [(1/5)t + 1/3] = 9
• 16 - (2x + 1) = (5x - 2) + 13

We already know how to solve one-step and two-step equations . We just extend the same process to solve the multi step equations as well. i.e., we use the properties of equations (like adding, subtracting, multiplying, or dividing both sides by some number/variable such that the equation is balanced) to solve the multi step equations. Let us see how to solve them.

Inverse Operations For Solving Multi-Step Equations

We solve the multi step equations by applying inverse operation on both sides to isolate the variable (making the variable alone on one side of the equation). Note that equality should not be disturbed when we apply any operation. For this, we should apply the same operation on both sides. For example, to solve x + 2 = 3 (of course, this is not a multi step equation), we should subtract 2 from both sides, then we get x + 2 - 2 = 3 - 2 which gives x = 1. Here since we are subtracting 2 from the left side, we should subtract the same number 2 from the right side as well. Wait! Why did we "subtract" 2? Because in the original equation x + 2 = 3, 2 was getting added to x, and to solve for x, we do NOT need + 2 on the left side, so we have subtracted it (as subtraction is the inverse operation of addition). Let us quickly revise the inverse operations:

• Addition and subtraction are inverse operations of each other
• Multiplication and division are inverse operations of each other
• Exponents and roots are inverse operations of each other Example: square and square root are inverse operations of each other, cube and cube root are inverse operations of each other, etc).

Applying the same operation on both sides without affecting the equality is proposed by the properties of equations. Here are some examples to understand them.

Solving Multi Step Equations

To solve multi step equations we may need to apply multiple types of inverse operations one after the other (that are mentioned in the previous section). The order of applying inverse operations is very important while solving multi step equations. For example, to solve the equation 2x + 4 = 6, the first step is NOT dividing both sides by 2, rather we subtract 4 from both sides. i.e.,

2x + 4 = 6 Subtracting 4 from both sides, 2x = 2 Dividing both sides by 2, x = 1

Our ultimate aim is to get just the variable on one side of the equation. We should aim at getting the answer something like "variable = something". Here are the important steps to solve multi step equations:

• Apply distributive property when you have a parenthesis.
• Combine like terms (if any).
• Collect like terms to one side of the equation. i.e., collect variable terms on the left side and the constants on the right side (or vice versa).
• Isolate the variable by inverse operations.

Here is an example to understand these steps.

Example: -2 (x - 3) - 7 = 7x + 11

This is a multi step equation with variable on both sides.

Applying distributive property (i.e., distributing -2 to the terms inside the brackets),

-2x + 6 - 7 = 7x + 11

Combing like terms (i.e., 6 - 7 = -1),

-2x - 1 = 7x + 11

Now our aim is to collect all x terms on the left and all constant on the right.

Subtracting 7x from both sides,

-9x - 1 = 11

Adding 1 on both sides,

Our aim is fulfilled now. Now, let us divide both side by -9,

Since x is isolated, it means that we have solved the equation.

Multi Step Equations with Fractions

Sometimes, multi step equations may involve one or more fractions in them. The easiest way of solving such equations is

• Find the LCD (Least Common Denominator) of all the denominators (of both left and right sides).
• Multiply every term on both sides of the equation by LCD .
• Apply inverse operations and isolate the variable.

Here is an example.

Example: Solve (1/4)t + 1/5 = (1/2)t + 5/3

The denominators are 4, 5, 2, and 3. Their LCD is 60. So multiply each term on both sides by 60.

(1/4)t × 60 + 1/5 × 60 = (1/2)t × 60 + 5/3 × 60

15t + 12 = 30t + 100

Now, the equation is free from fractions. We will proceed now. Subtracting 30t and 12 from both sides,

Dividing both sides by -15,

t = -88/15.

☛ Related Topics:

• Multi-Step Equation Calculator
• System of Equations Calculator
• Equation Calculator
• Two-Step Equations Calculator

Multi Step Equations Examples

Example 1: Solve the equation for "s": 3 (4s + 1) = 9.

We can solve the given multi step equation in two ways.

3 (4s + 1) = 9 Dividing both sides by 3, 4s + 1 = 3 Subtracting 1 from both sides, 4s = 2 Dividing both sides by 4, s = 1/2

3 (4s + 1) = 9 Distributing 3, 12s + 3 = 9 Subtracting 3 from both sides, 12s = 6 Dividing both sides by 12, s = 1/2

Answer: The solution is s = 1/2.

Example 2: Solve (1/3) x + 5 = 6x for x.

The given equation is:

(1/3) x + 5 = 6x

Multiply each term on both sides by 3 to avoid the fraction. x + 15 = 18x Let us collect the variables on one side. Subtracting x from both sides, 15 = 17x Dividing both sides by 17, x = 15/17

Answer: The solution of the given multi-step equation is x = 15/17.

Example 3: John is 5 years elder than his brother Michael. After 10 years, the sum of their ages is 35. Then how old is Michael now?

This is a multi-step equations word problem.

Let Michael is x years old. Then John's age = x + 5. After 10 years, the sum of their ages is 35. i.e.,

(x + 10) + (x + 5 + 10) = 35

Combining like terms ,

2x + 25 = 35

Subtracting 25 from both sides,

Dividing both sides by 2,

Answer: Michael is 5 years old.

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Multi Step Equations Questions

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FAQs on Multi Step Equations

How do you solve multi step equations.

To solve multi step equations , aim at getting the required variable on the left side of the equation. All the other numbers/variables should be removed from the left side by using inverse operations. The inverse operation of addition is subtraction (and vice versa) and the inverse operation of multiplication is division (and vice versa).

How to Solve Multistep Equations With Variables on Both Sides?

If a multi step equation has variables on both sides, then apply inverse operations to get the variable terms on one side and constant terms on the other side. Then solve for x. For example: 3x + 5 = 7x + 6 ⇒ 3x + 5 - 7x = 6 ⇒ -4x + 5 = 6 ⇒ -4x = 6 - 5 ⇒ -4x = 1 ⇒ x = -1/4.

How to Solve Multi Step Equations With Fractions?

If a multi step equation has fractions, then we can eliminate all fractions first by multiplying each term on both sides by LCD of all denominators. Then we can just apply the inverse operations and solve for the variable.

What are the 4 Steps of Multi Step Equations?

To solve multi step equations:

• Expand brackets by using distributive property if any.
• Combine like terms if any.
• Collect like terms of one type on either side.
• Apply inverse operations to isolate the variable.

Where to Find Multistep Equations Worksheets?

We can find multi step equations worksheets by clicking here . You can get varieties of problems (both equations and word problems) by clicking on the given link.

How to Solve Multi Step Equations Word Problems?

To solve multi step equations word problems:

• First, identify the variables with respect to the given context.
• Frame the multi step equation by reading the problem carefully.
• Solve it using the method that is explained on this page.

The multi step equation can also be solved directly through graph and it is often used in linear programming .

Solving Multi-Step Equations

In these lessons, we will learn how to

• Solve multi-step equations with whole numbers
• Solve multi-step equations with fractions
• Solve multi-step equations with decimals

Related Pages Solving Algebraic Equations More Algebra Lessons

Solving multi-step equations with whole numbers

The following diagram shows how to solve multi-step equations. Scroll down the page for more examples and solutions.

To solve a multi-step equation, we would start by trying to simplify the equation by combining like terms and using the distributive property whenever possible.

Consider the equation 2(x + 1) – x = 5. First, we will use the distributive property to remove the parenthesis and then we can combine like terms and then isolate the variable.

Example: Solve 2(x + 1) – x = 5

Solution: 2(x + 1) – x = 5 2x + 2 – x = 5 (use distributive property) x + 2 = 5 (combine like terms) x + 2 – 2 = 5 – 2 x = 3

How to solve multi-step equations by combining like terms and using the distributive property?

• 4x + 2x - 3x = 27
• 4a + 1 - a = 19
• 4(y - 1) = 36
• 16 = 2(x - 1) - x

Use distributive property to simplify multi-step equations

• 7(w + 20) - w = 5
• 9(x - 2) = 3x + 3
• Lydia inherited a sum of money. She split it into five equal chunks. She invested three parts of the money in a high interest bank account which added 10% to the value. She placed the rest of her inheritance plus $500 in the stock market but lost 20% on that money. If the two accounts end up exactly the same amount of money in them, how much did she inherit?

Solving Multi-Step Equations With Fractions

To solve an equation with fractions, we first try to change it into an equation without fractions. Then, we can solve it using the methods we already know.

How to solve multi-step equations with fractions?

• 1/4 x + 3 = 2
• 1/2 (k - 8) = 6
• 1/2 d + 2 = 3/4
• -3/4 x + 1/4 = 1/2

How to solve Multi-Step Equations with Fractions & Decimals?

• 3/2 n - 6 = 22
• 2/5 x + 2 = 3/4
• 0.035m + 9.95 = 12.75

Solving Multi-Step Equations With Decimals

The steps involved in solving multi-step equations with decimals are the same as those in equations with whole numbers. The complication may lie more in the multiplication and division of decimals rather than the steps. Another method would be to multiply each term of the equation by ten (or hundred) to convert the decimals to whole numbers and then solve the equation.

How to solve multi-step equations with decimals?

• 0.4x + 9.2 = 10
• 0.4(a + 2) = 2
• 1.2c + 2.6c = 4.56

How to solve multiple step linear equations involving decimals?

• Remove parentheses by using the distributive property. Then combine like terms on each side.
• Add or subtract, as needed, to get all variable terms on one side and all constant terms on the other. Then combine like terms.
• Multiply or divide to solve for the variable.
• Check all possible solutions.

Examples: Solve each equation.

• 1.2x - 5.12 - 0.9x = 1.6
• 5x - 0.2(x - 4.2) = 1.8

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