lesson 2 homework practice write two step equations

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Two Step Equations Worksheets

The first major leap forward for students in prealgebra is working with equations that require multiple steps to solve. When an equation is presented to you, it is balanced meaning that both sides are equal. You can easily rearrange any equation by simply performing the operation of your choice. You just need to make sure to process that operation on both sides of the equal symbol. Whatever you decide to do to one side, will be done to the other. In most cases we are simply trying to get a variable by itself and determine what it is equal to. The most commonly used variable is the letter “x”. These worksheets and lessons help students learn how to manage problems that involve solving equations in at least two steps.

Aligned Standard: 7.EE.B.3

• Solve For X Step-by-Step Lesson - Two thumbs up if you get this one. You get two either way!
• Guided Lesson - Play through on these. Make sure to get rid of the constant first.
• Guided Lesson Explanation - In some cases, I made two steps into three steps.
• Independent Practice - I used all numbers under 20 in the operations to make sure basic operations don't trip you up.
• Matching Worksheet - Find the value of the variables in all cases.
• Practice Worksheet Pack - You will find five practice worksheets here.
• Write the Equation Worksheet Pack - We give you the words. We want the integers in return.
• Two Step Equations in Words Lesson Practice Sheet - This one was written well. I think I'm on version 30 of this.
• Two Step Equations in Words Worksheet - Now you not only need to write the equation, but solve it too.
• Answer Keys - These are for all the unlocked materials above.

Homework Sheets

Start by getting x by itself. The next step is to take a sentence and turn into a math sentence that is then solvable.

• Homework 1 - In order to get x on the left side we have to remove 4 using addition.
• Homework 2 - Write the sentence as an equation and solve it. Four less than a number divided by 3 is six.
• Homework 3 - For the first step, in order to get x on the left side we have to remove 12 using subtraction.

Practice Worksheets

The sentence based problems really give many students trouble.

• Practice 1 - It is always good to start with numbers that have no variables attached to them.
• Practice 2 - In the second step we will remove the coefficient using division. The opposite of multiplication is division.
• Practice 3 - Write the sentences as equations and solve.

Math Skill Quizzes

Remind students to combine like terms as much as possible and then proceed from there.

• Quiz 1 - Write the sentences as equations and solve. You are basically trying to represent what is presented with numbers and symbols.
• Quiz 2 - Solve for p: 6p + 2p = 72. There are some single steps in here too to allow for some confidence.
• Quiz 3 - You will find you deal a lot with negative values here. Just remember to counter the operations.

How to Approach Two-Step Equations

When you start to learn algebra, it gets more complicated every step of the way. Nonetheless, after learning a single-step equation in which you only have to switch numbers to the other side of the relational operator. For instance, when you have an equation like x + 3 = 0, you need to subtract 3 (since it's the same number on the left-hand-side) on both sides to have an equation x + 3 - 3 = 0 - 3. The answer will be x = -3. However, to solve the two-step equation, we have to learn one more single step. Let's learn it in the form of an example.

For example, we will take the equation 3x + 5 = 65. The entire goal is to get the variable (x) by itself.

The first step, in this case, will be to subtract 5 from both sides of the relational sign (3x + 5 – 5) = (65 – 5).

The result is 3x = 60; however, you haven't found the value of x alone. There is a coefficient attached to x.

In such circumstances, you need to divide with the same number as the coefficient attached to the variable. Therefore, we will divide both sides with 3. 3x / 3 = 60 / 3. So x = 20.

It is all about determining what operations are in play and then just drop the opposites to cancel them out, as you have seen in our example.

The Two Steps You Need to Solve These Problems

These problems can seem much more reasonable when you put them into perspective and think in this manner when tackling them:

1) Start by countering the math operations that are presented to you. Just remember that addition and subtraction will counter one another. Multiplication and division are adversaries as well. Whatever operation you choose to use to create balance, just use the same operation on both sides of the equation and you are all set. This will help you tip the scales in your favor.

2) The last step is just to get the variable by itself. This will again require you to counter the operations that are in place, and you are good to go.

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Writing Real-World Two-Step Equations Homework/Practice Worksheet

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Description

This resource can be used as in-class practice, homework, or an additional worksheet for reinforcement of writing two-step equations. All 8 examples are real-world problems and instructions prompt students to define variables, write and solve a two-step equation, and explain the solution in a sentence. These instructions are pulled right from the chart on my Writing Two-Step Equations from Word Problems Guided Notes .

A teacher copy of the worksheet is included. The practice/worksheet can be hole punched and placed right into a student's binder or copied at 70 - 75% to create a size that would fit into a composition sized interactive notebook.

Check out my other middle school expressions and equations resources:

Two-Step Equations Notes and Practice Bundle

Writing Equivalent Expressions Guided Notes

Evaluating Expressions Guided Notes

Introduction to Algebra Guided Notes

One Step Equations Maze

Two-Step Equations BOOM Cards

Two-Step Equations Matching Activity Puzzles

Combining Like Terms BOOM Cards

Combining Like Terms Matching Activity Puzzles

Combining Like Terms (Simplifying Expressions) Guided Notes Bundle

7th Grade Expressions Guided Notes Bundle

Simplifying Expressions Foldable - Distributive Property and Combining Like Terms

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Two-Step Equations

Solving two-step equations.

There is no doubt that solving a two-step equation is extremely easy. As the name implies, two-step equations can be solved in just two steps. If this is your first encounter with two-step equations, don’t worry because we will go over enough examples to make you familiar with the process.

When solving an equation in general, we always keep in mind the notion that whatever we do to one side of the equation should also be done to the other side to ensure that the equation remains balanced.

We know that we have completely solved a two-step equation if the variable, usually represented by a letter in an alphabet, is isolated on one side of the equation (either left or right) and the number is located on the opposite side.

The USUAL way of solving a two-step equation:

Note : This is the “usual” method because most of the two-step equations are solved this way. Notice that Step 2 can alternatively be replaced by Step 3 which are the same essentially.

1)  First, add or subtract both sides of the linear equation by the same number.

2)  Secondly, multiply or divide both sides of the linear equation by the same number.

3)* Instead of step #2, always multiply both sides of the equation by the reciprocal of the coefficient of the variable.

Examples of How to Solve Two-Step Equations

Example 1: Solve the two-step equation below.

As the name of this linear equation suggests, it requires two steps in order to solve for the unknown variable. Generally, the first step involves getting rid of the number “farthest” from the term with a variable being solved. Then, we eliminate the number “closest” to the variable. The number is either multiplying or dividing the variable. It is also called the coefficient of the term.

The variable here is [latex]x[/latex]. Our goal is to solve [latex]x[/latex] by isolating it on one side of the equation. Keeping the variable on the left or on the right doesn’t make any difference. It is up to you! In this problem, let’s keep it on the left side since it’s already there.

On the side (left side of the linear equation) where the variable is located, notice that [latex]2[/latex] is “closest” to variable [latex]x[/latex], and [latex]5[/latex] is “farthest” one.

This simple observation allows us to decide which number to eliminate first. It is obviously [latex]+5[/latex] because it is farther between the two. The opposite of [latex]+5[/latex] is [latex]-5[/latex], that means we will subtract both sides of the equation by [latex]5[/latex].

After eliminating [latex]5[/latex] on the left side of the equation by subtracting both sides by [latex]5[/latex], it’s time to get rid of the number closest or directly attached to [latex]x[/latex] which is [latex]2[/latex] in [latex]2x[/latex]. Since [latex]2[/latex] is multiplying the variable [latex]x[/latex], its opposite operation is to divide by [latex]2[/latex].

After dividing both sides by [latex]2[/latex], we obtain the final answer or solution to the given two-step linear equation.

Just a quick reminder, it is considered solved because the coefficient of the variable is just  positive one, [latex]+1[/latex].

Example 2: Solve the two-step equation below.

Our goal is to keep the variable [latex]x[/latex] on one side of the equation. It doesn’t matter which side, however it is a “standard” practice to keep the variable being solved to the left side. Some algebra teachers may require you to keep the variable to the left and there’s nothing with that. Personally, I don’t mind where you keep the variable, either left or right, as long as the isolated variable on one side of the equation has a coefficient of [latex]+1[/latex].

The first step involves removing the number “farthest” from variable [latex]x[/latex]. Notice that [latex]-3[/latex] is “closest” to [latex]x[/latex], while [latex]-8[/latex] is “farther away”. So then, we can eliminate [latex]-8[/latex] by adding to its opposite which is [latex]+8[/latex].

The second step involves getting rid of the number closest to the variable [latex]x[/latex] which is [latex]-3[/latex]. Since [latex]-3[/latex] is multiplying the variable x, its opposite operation is to divide by [latex]-3[/latex]. After dividing both sides by [latex]-3[/latex], we have solved the linear equation.

Quick reminder, [latex]-3[/latex] divided by [latex]-3[/latex] is equal to [latex]+1[/latex].

Example 3: Solve the two-step equation below.

Here’s a situation where we can isolate the variable [latex]x[/latex] to the right side of the equation since it is already there.

Looking at the right side of the equation where the variable is located, the number [latex]3[/latex] is closest to [latex]x[/latex] because [latex]3[/latex] is dividing the variable [latex]x[/latex]. On the other hand, the number [latex]26[/latex] is “farther away”. This implies that we will have to deal with [latex]+26[/latex] by subtracting both sides of the equation by [latex]26[/latex]. The reason we subtract is that the additive inverse of [latex]+26[/latex] is [latex]-26[/latex].

The second step is to get rid of the denominator [latex]3[/latex]. Since [latex]3[/latex] is dividing the [latex]x[/latex], its opposite operation is to multiply by [latex]3[/latex].

After multiplying both sides by [latex]3[/latex], we have arrived at the final answer. You may rewrite your final answer as [latex]x = -9[/latex].

Example 4: Solve the two-equation below.

This may appear to be a multi-step equation, but it is not. It can be solved in two steps. Don’t be bothered by the fractions because they are very easy to deal with. In this case, you will apply the rule of fraction addition. The rule states that if you are adding two fractions that have the same denominator, just add the numerators then copy the common denominator.

Back to solving the two-step equation above, to remove the fraction on the left side which is [latex]\Large{ – {3 \over {10}}}[/latex], we will add [latex]\Large{{3 \over {10}}}[/latex] to both sides of the equation.

The reason that we are adding instead of subtracting is because the additive inverse of [latex]\Large{ – {3 \over {10}}}[/latex] is [latex]\Large{+ {3 \over {10}}}[/latex]  .

After adding [latex]\Large{{3 \over {10}}}[/latex] to both sides, only [latex]{\Large{{2 \over 5}}}x[/latex] remains on the left side.

For the right side of the equation, we have [latex]\Large{{9 \over {10}} + {3 \over {10}} = {{12} \over {10}}}[/latex].

Everything I stated above is just the first step. Now, moving on to the second step. Look at the coefficient of the variable [latex]x[/latex]. It is [latex]\Large{{2 \over 5}}[/latex] that means its reciprocal is [latex]\Large{{5 \over 2}}[/latex].

To finally solve the given equation, we will multiply both sides of the equation by the reciprocal of the coefficient of the variable in question. Here’s the complete step-by-step solution:

Take a quiz:

Two-Step Equations Quiz

You may also be interested in these related math lessons or tutorials:

Two-Step Equations Practice Problems with Answers

Multi-Step Equations Practice Problems with Answers

Two-step Equations – Definition, Steps, Facts, Examples, FAQs

What are two-step equations, how to solve two-step equations, what are the rules to solve two-step equations, solved examples on two-step equations, practice problems on two-step equations, frequently asked questions on two-step equations.

Two-step equations are mathematical equations that require only two steps to solve and find the value of the variable. In order to solve two step equations, we need to undo two operations.

Two-step equation example:

Two-Step Equations: Definition

Two-step equations are algebraic equations that can be solved in two steps.

Two step equations may come in different forms involving combinations of addition , subtraction , multiplication , or division operation.

Examples of two-step equations:

• $2x + 1 = 5$
• $\frac{7x}{\;-\;2} = 21$

Related Worksheets

When solving two-step algebraic equations, the goal is to isolate the variable on one side of the equation. 

To isolate the variable, we have to undo the involved operations by using their inverse operations (opposite operations) and solve for the varibale. 

In most of the two-step equations, usually we first use addition or subtraction operation to isolate the variable and in the second step, we use multiplication or division to find the value of the variable.

Example: Solve $2x + 1 = 5$

Subtract 1 from both sides.

$2x + 1\;-\;1 = 5\;-\;1$

Divide both sides by 2.

$\frac{2x}{2} = \frac{4}{2}$

$x = 2$                        

To solve two step equation, we need to balance both sides of the equation using following rules: 

• Undo the addition by subtracting both sides with the same number.
• Undo subtraction by adding both sides with the same number.
• Undo the multiplication by dividing both sides with the same number.
• Undo the division by multiplying the same number to both sides. 

Facts about Two-step Equations

• Solving two-step equations helps build the foundation for solving multi-step algebraic equations.
• The first step in solving a two-step equation is usually undoing addition or subtraction by applying the inverse operation.
• The second step generally involves undoing multiplication or division by applying the inverse operation.

Conclusion 

Two-step equation has one variable and two mathematical operations in it. Using the order of operations in reverse we can reach the solution by finding the value of an unknown variable. Even though we can use different methods to solve this two step equation, solving through reverse order of operations is the easiest way.

Example 1: Solve 3x + 12 = 21.

Solution:  

 3x + 12 = 21

Undo the addition operation by subtracting 12 from both sides of the equation. 

3x + 12 – 12 = 21 – 12 

3x + 0 = 9 

Undo the multiplication operation by dividing both sides of the equation by 3 we get,

$\frac{3}{3} x = \frac{9}{3}$

 x = 3

Example 2: Sam sold one-fourth of his watch collection, and then sold 5 more watches. How many watches did Sam have in the beginning if he sold a total of 10 watches?  

Solution: 

Suppose that Sam had x number of watches initially in his collection.

First, he sold one-fourth of his collection, which is $\frac{x}{4}$. 

Next, he sold 5 more watches.

So, total number of watched he sold $= \frac{x}{4} + 5$

It is given that Sam sold a total of 10 watches.

Thus,  $\frac{x}{4} + 5 = 10$

Undo the subtraction by adding the number 5 on both sides of the equation we get, 

$\frac{x}{4} + 5 \;-\; 5 = 10 \;-\; 5$

$\frac{x}{4} = 5$

Undo the division operation by multiplying number 4 on both sides of the equation we get, 

$\frac{4x}{4} = 5 \times 4$

Hence, Ramsey had 20 watches with him in the beginning. 

Example 3: Solve 2 (x + 7) = 16 .

2 (x + 7) = 16

$\frac{2}{2} (x + 7) = \frac{16}{2}$ divide by 2 on both sides

x + 7 – 7 = 8 – 7 subtract 7 from both sides 

x = 1                                                                                                                                                                                 

Another way: If we expand this equation, we get

 2x + 14 = 16

2x = 16 – 14

Example 4: Solve the two step equation $\frac{x \;-\; 2}{3} = 1$

$\frac{x \;-\; 2}{3} = 1$ 

$\frac{3(x \;-\; 2)}{3} = 1 \times 3$ multiplying 3 on both sides of the equation we get,                                                

x – 2 = 3 adding 2 on both sides of the equation we get, 

x – 2 + 2  = 3 + 2 

Two-step Equations - Definition, Steps, Facts, Examples, FAQs

Attend this quiz & Test your knowledge.

Find the value of x if 2x+3=5.

Solve: - x + 1 = 12, what are two-step equations.

What is a multi-step equation?

A multi-step equation contains mixed operations such as addition, subtraction, multiplication, or division. It requires two or more steps to solve the equation. We can use the reverse order of operations to solve them.

What are one-step equations?

One-step equations are algebraic equations that require only one step to solve.

How do you check if the solution to a two-step equation is correct?

To check the solution, substitute the found value back into the original equation and verify if both sides are equal.

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Solving Two-Step Equations

Order Matters

What does the order of operations have to do with solving equations? Quite a bit, as it turns out. In this lesson, Kate uses the order of operations to explain why we solve two-step equations the way we do. 

STEP 1: WATCH THE VIRTUAL CLASS VIDEO AND TAKE NOTES

Make sure to note the following:.

Simplifying and solving are opposite procedures. Because of this, you actually work the order of operations BACKWARDS when solving equations. 

Ask a Question

Do you have a question about the class video, practice, or example problems? Post a picture to our FB group for immediate assistance. Be sure to include the lesson name and level in your post. 

STEP 2: COMPLETE ONE LEVEL OF PRACTICE

Solve two-step equations involving whole numbers and the basic operations.

EXPERIENCED

This worksheet from Kuta Software has you solve two step equations with negatives. 

Mixed practice with negatives, fractions, and exponents. 

NEED MORE HELP? CHECK OUT THE EXAMPLE VIDEOS 

Beginner Solution Playlist

Experienced Solution Playlist

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Chapter 4, Lesson 5: Solving Two-Step Equations

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