solving two step equations assignment quizlet

Solving Two-Step Equations: Explanations, Review, and Examples

• The Albert Team
• Last Updated On: February 16, 2023

So, we’ve made it past one-step equations! Woo-hoo!

Our reward? Solving two-step equations !

Don’t worry: whether this is your first experience with two-step equations, or you are reviewing for an exam, this blog will guide you through defining two-step equations, examples of two-step equations, and how to solve two-step equations (including fractions and word problems). Let’s get started!

What We Review

What is a two-step equation?

Remember, an equation is a mathematical sentence that uses an equal sign, = , to show that two expressions are equal.

Very similar to one-step equations , a two-step equation is an equation that only requires two steps to solve. We will use a mix of addition, subtraction, multiplication, and division to solve these equations. 

Examples of two-step equations

Two-step equations come in many types. You might have some equations that require subtraction, then division to solve, or an equation that requires multiplication, then division to solve.

Here are some examples of two-step equations:

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How to solve two-step equations

Remember: To solve equations, we must use inverse operations to isolate the variable. Examples of inverse operations are: 

We must first eliminate any constants from the side of the equation with the variable. Additionally, whatever we do to one side of the equation, we must also do to the other. Here are two examples of how to solve two-step equations: 

First, let’s solve for x in the following equation: 

To check your answer, you can substitute 4 into the variable to see if the equation is true:

Thus, x = 4 is the correct solution.

Now, we can trying solving for y in the following equation: 

To check you answer, you can simplify substitute 7 into the variable to see if the equation is true:

Thus, y = 7 is the correct solution.

For more, watch the video from mathantics below showing how the solve 2-step equations:

How to solve two-step equations with fractions

Unfortunately, equations do not always contain only whole numbers. Never fear! We can still solve two-step equations even when fractions are involved. 

Here is an example of solving a two-step equation with a fraction: 

Solve for m in the following equation:

To check you answer, you can simplify substitute 9 into the variable to see if the equation is true: 

Thus, m = 9 is the correct solution.

Is there a way to make solving two-step equations with fractions easier? I’m glad you asked! If you want to eliminate fractions completely when solving a two-step equation, you can simply multiply the whole equation by the Least Common Denominator . Here is an example showing this method: 

Solve for x in the following equation: 

Since the denominators are 2, 3, \text{and } 6 , the least common denominator would be 6 . Therefore, to eliminate all fractions from the problem, we would multiply each term by 6 .

To check you answer, you can simplify substitute 1 into the variable to see if the equation is true: 

Here’s a video from Brian McLogan on how to solve two-step equations with fractions:

Two-step equation word problems

Similar to One-Step Equations , we can model real-life scenarios with two-step equations. Once we model the situation with an equation, we simply solve as we have above. 

For instance, model the following situations with an equation and find a solution that makes the situation true. 

To solve for c , we will first subtract 10 from each side: 

Then to find the cost of one ticket, we will divide each side by 3  

To solve for t , we must first multiply each side by 4 to eliminate the denominator

Solving Two-Step Equations: Keys to Remember

Remember, just like solving One-Step Equations there are some key facts to remember: 

• A two-step equation is an equation that requires two steps to solve
• We must eliminate any constant that is on the same side as the variable first
• To solve, use the inverse operations to isolate the variable by itself
• Remember whatever you do to one side, you must do to the other
• To check the solution, simply substitute the value into the variable to see if the equation is true
• You can model real-life situations with an equation and solve for a correct solution

Read these other helpful posts:

• Solving One-Step Equations
• Solving Multi-Step Equations
• Forms of Linear Equations
• View ALL Algebra 1 Review Guides

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Two-Step Equations

Two step equations are equations that can be solved within exactly two steps. Two step equations are extremely easy to solve. As the name suggests, two step equations take only two steps to solve. These equations are just a little complicated than the one step equations. While solving a two step equation, we need to perform the operation on both sides of the equals to sign.

In this article, we will understand the meaning of two step equations with integers, decimals, and fractions, how to solve them, the golden rule to solve two step equations along with some examples for a better understanding.

What are Two Step Equations?

Two step equations are algebraic problems that take just two steps to solve. The two step equation is a linear equation in one variable . While performing an operation for solving a two step equation, we need to perform the same operation on both sides of the equation. We isolate the variable on one side of the equation to determine its value.

Two Step Equations Definition

Two step equations are algebraic equations and are the equations that can be solved in exactly two steps and gives the final value of the variable in two steps. Generally, two step equations are of the form ax + b = c, where a, b, c are real numbers. A few examples of two step equations are:

• 0.3y + 5 = 1
• (2/3)z - 12 = 10

Solving Two Step Equations

Two step equations are very easy to solve. It includes just one extra step as compared to one-step equations to solve. We can solve a two-step equation by isolating the variable (usually represented by an alphabet or letter) on one side of the equation and all other values on the other side. The general two steps to solve the two-step equations are:

• Step 1: Addition and subtraction to isolate the variable.
• Step 2: Multiplication  or division  to determine the value of the variable.

Let us consider a few examples and solve two-step equations to understand the concept of solving two-step equations.

Example 1: Solve the equation 2x + 6 = 12.

To solve the two step equation 2x + 6 = 12, we need to determine the value of x. Let us solve it step-wise.

Step 1: Subtract -6 from both sides of the equation to isolate the variable x.

2x + 6 - 6 = 12 - 6

Step 2: Divide both sides of the equation by 2 to solve for x.

Hence, we have solved the equation 2x + 3 = 12 in just two steps.

Thus the two-step equation can be easily solved in a sequence of steps, as presented above.

Two-Step Equations with Decimals and Fractions

Two step equations that have decimals and fractions as the coefficient of the variable and constant term are said to be two step equations with decimals and fractions. A few examples of two step equations with fractions and decimals are:

• 0.3 x + 2/3 = 1
• 3x - 0.5 = 1.2
• (1/3) x + 4/5 = 3/4

These equations are solved in the same manner as the general two steps equations and the same steps are followed to determine the value of the variable.

Golden Rule to Solve Two Step Equations

The golden rule to solve two step equations is to perform all operations simultaneously on both sides of the equation. To isolate the variable on one side of the equation and to determine its value, we first add or subtract on both sides of the equation and then multiply or divide on both sides to get the final solution of the two step equation.

Important Notes on Two Step Equations

• Remove the parentheses and combine like terms to simplify each side of the two-step equation.
• Always remove the constant first by adding or subtracting the appropriate number.
• Always verify the solution in the end.

Topics Related to Two Step Equations

• Equations in Math
• Simple equations
• Algebraic formulas

Two Step Equations Examples

Example 1: Solve the two step equation (x/6) - 7 = 11

Solution: To solve the given two step equation, we will follow the steps discussed above in the article.

Step 1: Add 7 to both sides of the given two step equation

(x/6) - 7 + 7 = 11 + 7

⇒ (x/6) = 18

Step 2: Multiply both sides of the equation by 6.

6 × x/6 = 6 × 18

Answer: Hence the solution to the given two step equation (x/6) - 7 = 11 is x = 108.

Example 2: Determine the solution of the two step equation (2/3) z + 0.8 = 1.5

Step 1: Subtract 0.8 from both sides of the given two step equation

(2/3) z + 0.8 - 0.8 = 1.5 - 0.8

⇒ (2/3) z = 0.7

Step 2: Multiply both sides of the equation by (3/2).

(3/2) × (2/3) z = (3/2) × 0.7

Answer: Hence the solution to the given two step equation (2/3) z + 0.8 = 1.5 is x = 1.05

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Practice Questions on Two Step Equations

Faqs on two step equations, what are two step equations in algebra.

Two step equations are algebraic equations that take just two steps to solve.First, the variable is isolated by adding or subtracting a numeric value on both sides of the equation. Secondly, the value of the variable is computed by multiplying or dividing the variable by an appropriate number.

What are the Steps to Solve Two Step Equations?

The general two steps to solve two step equations are:

• Step 1: Simplify the given equation by removing all brackets and parenthesis:
• Step 2: Add or subtract to isolate the variable.
• Step 2: Multiply or divide to determine the value of the variable.
• Step: Verify the answer by substituting it in the given equation .

How to Solve Two Step Equations?

Two step equations can be solved by following two quick steps:

• Step 1: Add or subtract numbers on either sides, to isolate the variable.

IS Two Step Equation the Same as A Multi-Step Equation?

Two-step equation can also be called a multi-step equation since it involves more than one step. And a multi-step equation can have two or more steps, in the process of solving the equation.

What is the Difference Between One Step and Two Step Equations?

One step equations take just one step to solve whereas two steps equations take two steps to get to the solution. Two-step equations include just one extra step as compared to one step equations to solve.

What is the Goal of Solving Two Step Equations?

The goal of solving two step equations is to isolate the variable and determine the value of the variable. And in the end, the variable should satisfy the given two step equation.

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Two-step linear equations.

• Solving a Two-Step Linear Equation

1. Solving a Two-Step Linear Equation

This lesson builds upon what we learned earlier about using inverse operations to isolate the variable, applying this knowledge to equations requiring two steps to solve for the unknown value. Remember, when solving an equation for a variable, our main goal is to isolate a variable. In other words, we want to get the variable by itself on one side of the equation, with all other expressions on the other side of the equals sign. In this process, we must always remember that if we perform an operation on one side of the equation, we must do the same on the other side of the equal sign. Let's look at an example:

hint To solve a two-step equation, first move the constants (numbers that don’t have variables) to one side of the equation, then isolate the variable by canceling out the coefficient (numbers multiplied or divided by the variable).

big idea Whatever we do on one side of the equation has to be done on the other side of the equation. This is known as the Rule of Equality.

IN CONTEXT When working with word problems, it is always a good idea to start by clearly labeling the variables in a short list before we begin to solve the problem. This is important in all word problems involving variables, not just consecutive numbers or geometry problems. This is shown in the following example: Speedy Taxi company charges $5 for the initial pick up and then $2/mile. If you spent $33, how many miles did you travel? Now set up your equation. Solve for x . Subtract 5 from both sides because it is the opposite of addition (or, in this case, a positive number). This is the result after subtracting 5 from both sides. Divide both sides by 2 because it is the opposite of multiplication. Our Solution: You traveled 14 miles.

IN CONTEXT A sofa and a loveseat together costs $444. The sofa costs double the loveseat. How much do they each cost? With no information about the loveseat, this is our x . Sofa is double the loveseat, so we multiply by 2. Together, they cost $444, so we add them. Replace sofa and loveseat with labeled values. Parentheses are not needed; combine like terms x and 2 x . Divide both sides by 3. Our solution for x . Replace x with 148 in the original list. The loveseat costs $148 and the sofa costs $296.

hint Be careful of problems such as these. Many students see the phrase "double" and believe that it means we only have to divide 444 by 2 and get sofa equals 222 for one or both of the prices. As you can see, this will not work. By clearly labeling the variables in the original list, we know exactly how to set up and solve these problems.

summary The process of solving a two-step linear equation involves isolating the variable you want to solve for. When isolating a variable, it is helpful to have a review of inverse operations: addition and subtraction are inverse, and multiplication and division are inverse. Keep in mind when applying inverse operation that this will cancel the operations around the variable. Also, in using the inverse operations, use the order of operations in reverse order. Finally, simplifying before isolating a variable, such as distributing or combining like terms, can be helpful. These types of equations strengthen your problem solving skill by enabling you to focus on complex ideas, follow multi-step processes, and think critically to find solutions. Best of luck in your learning!

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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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• 1. Multiple Choice Edit 1.5 minutes 1 pt Solve the equation: -5x = 25 x = 20 x = -20 x = -5 x = 5
• 2. Multiple Choice Edit 1.5 minutes 1 pt 19  + a  =  -30 a = -570 a =  -49 a = -11 a = -30/19
• 3. Multiple Choice Edit 1.5 minutes 1 pt x - 8 = -5  x = 13 x = -13 x = 3 x = -3
• 4. Multiple Choice Edit 1.5 minutes 1 pt 6b = 30 b = 36 b = 24 b = 180 b = 5
• 5. Multiple Choice Edit 1.5 minutes 1 pt  16 - 4y = 48 y = 16 y = -8 y = -28 y = -16
• 7. Multiple Choice Edit 1.5 minutes 1 pt Which equation has x = 5 as the solution? x + 15 = 10 2x = 5 2x = 10 -5x=25
• 8. Multiple Choice Edit 1.5 minutes 1 pt -16+ x= -15 x = 2 x = 0 x = -1 x = 1
• 9. Multiple Choice Edit 1.5 minutes 1 pt -7b+ 35 = 49 b = 7 b = -7 b = 2 b = -2
• 11. Multiple Choice Edit 1.5 minutes 1 pt 144=12z z = 20 z = 11 z = 12 z = 132
• 12. Multiple Choice Edit 1.5 minutes 1 pt What is the inverse operation needed to solve for p? 765 = p - 254 subtraction addition multiplication division
• 13. Multiple Choice Edit 1.5 minutes 1 pt Solve the equation: x - 5.5 = 10.5 x = 16 x = -15 x = 5 x = -5
• 14. Multiple Choice Edit 1.5 minutes 1 pt Solve 2x - 3 = 1  x = 3 x = 2 x = 1 x = -2
• 15. Multiple Choice Edit 1.5 minutes 1 pt Solve  3x - 7 = 32 x = 13 x = -13 x = 12 x = -12
• 16. Multiple Choice Edit 1.5 minutes 1 pt Solve   -5x + 9 = 24 x = -3 x = 3 x = -6.6 x = -12
• 17. Multiple Choice Edit 1.5 minutes 1 pt Solve   -3y + 7  = 22 y = -5 y = 5 y = 15 y = -15
• 18. Multiple Choice Edit 1.5 minutes 1 pt In the equation 4y +-9  = -36 y = -4 True False
• 19. Multiple Choice Edit 1.5 minutes 1 pt The steps in solving 5b - 6 = 24 are:  Add 6 to both sides and divide both sides by 5 Add 6 to both sides and multiply both sides by 5 Subtract 6 from both sides then multiply both sides by 5 Subtract 6 from both sides then divide both sides by 5
• 20. Multiple Choice Edit 1.5 minutes 1 pt What does INVERSE OPERATION mean? Operate Seven Opposite Mrs. Asher's Favorite word!

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