Module 4 Lesson 3

Module 4

Eureka Math - 8th Grade

Lesson 3

03

Linear Equations in x

Linear Expressions and Equations

They can be written as a

linear equation in x

When two linear

expressions are equal…

Then x is capable of having
a value (or multiple, or even

infinite values)

In other words…

If a linear equation is true…

Solving Equations

Is the Same as Solving for X

Is the Solution of

the Equation

The Value of x

Is 4+15x=49 True?
Yes. Because if you

simplify, you’ll find that x=3

Example

Problem 1

○ Is the equation a true statement when 𝑥 = −3?

○ In other words, is −3 a solution to the equation 6𝑥 + 5 =

5𝑥 + 8 + 2𝑥?

○ Explain.

Problem 1

○Is the equation a true statement when 𝑥 = −3?

○In other words, is −3 a solution to the equation 6𝑥 + 5 = 5𝑥 + 8 + 2𝑥?

○Explain.

6𝑥 + 5 = 5𝑥 + 8 + 2𝑥

Step 1: Substitute x with -3

Problem 1

○Is the equation a true statement when 𝑥 = −3?

○In other words, is −3 a solution to the equation 6𝑥 + 5 = 5𝑥 + 8 + 2𝑥?

○Explain.

6𝑥 + 5 = 5𝑥 + 8 + 2𝑥

6 (-3) + 5 = 5 (-3) + 8 + 2 (-3)

Step 2: Simplify Using Order of Operations

Problem 1

○Is the equation a true statement when 𝑥 = −3?

○In other words, is −3 a solution to the equation 6𝑥 + 5 = 5𝑥 + 8 + 2𝑥?

○Explain.

6𝑥 + 5 = 5𝑥 + 8 + 2𝑥

6 (-3) + 5 = 5 (-3) + 8 + 2 (-3)

-18 + 5 = -15 + 8 + (-6)

Step 3: Keep Simplify Using Order of Operations

Problem 1

○Is the equation a true statement when 𝑥 = −3?

○In other words, is −3 a solution to the equation 6𝑥 + 5 = 5𝑥 + 8 + 2𝑥? Explain.

6𝑥 + 5 = 5𝑥 + 8 + 2𝑥

6 (-3) + 5 = 5 (-3) + 8 + 2 (-3)

-18 + 5 = -15 + 8 + (-6)

-13 = -7 + (-6)

Step 4: Determine if the equation is a true statement

Multiple Choice

Is −3 a solution to the equation 6𝑥 + 5 = 5𝑥 + 8 + 2𝑥?

• Yes.

  -3 is a solution for the equation.

• No.

  -3 is not a solution for the equation.

Problem 2

○ Does 𝑥 = 12 satisfy the equation 16 − 1 2 𝑥 = 3 4 𝑥 + 1?

○ Explain.

Problem 2

○Does 𝑥 = 12 satisfy the equation 16 − 1 2 𝑥 = 3 4 𝑥 + 1? Explain.

16 - 12x = 34x + 1

Step 1: Substitute the value of x

Problem 2

○Does 𝑥 = 12 satisfy the equation 16 − 1 2 𝑥 = 3 4 𝑥 + 1? Explain.

16 - 12x = 34x + 1

12 - 12 (12) = 34 (12) + 1

Step 2: Simplify Using the Order of Operations

Problem 2

○Does 𝑥 = 12 satisfy the equation 16 − 1 2 𝑥 = 3 4 𝑥 + 1? Explain.

16 - 12x = 34x + 1

12 - 12 (12) = 34 (12) + 1

12 - 144 = 408 + 1

Step 3: Keep Simplifying Using the Order of Operations

Problem 2

○Does 𝑥 = 12 satisfy the equation 16 − 1 2 𝑥 = 3 4 𝑥 + 1? Explain.

16 - 12x = 34x + 1

12 - 12 (12) = 34 (12) + 1

12 - 144 = 408 + 1

132 = 409

Step 4: Determine if it’s possible for x to equal 12.

Multiple Choice

Does 𝑥 = 12 satisfy the equation 16 − 1 2 𝑥 = 3 4 𝑥 + 1?

• Yes, x=12 does satisfy the equation.

• No, x=12 does not satisfy the equation.

Problem 3

○ Chad solved the equation 24𝑥 + 4 + 2𝑥 = 3(10𝑥 − 1) and

is claiming that 𝑥 = 2 makes the equation true.

○ Is Chad correct?

○ Explain.

Problem 3

○Chad solved the equation 24𝑥 + 4 + 2𝑥 = 3(10𝑥 − 1) and is claiming that 𝑥 = 2 makes the

equation true.

○Is Chad correct? Explain.

24x + 4 + 2x = 3 (10x -1)

Step 1: Substitute the value of x

Problem 3

○Chad solved the equation 24𝑥 + 4 + 2𝑥 = 3(10𝑥 − 1) and is claiming that 𝑥 = 2 makes the

equation true.

○Is Chad correct? Explain.

24x + 4 + 2x = 3 (10x -1)

24 (2) + 4 + 2 (2) = 3 (10 (2) - 1)

Step 2: Simplify Using the Order of Operations

Multiple Choice

Given the equation 24𝑥 + 4 + 2𝑥 = 3(10𝑥 − 1)

Does 𝑥 = 2 makes the equation true?

• Yes, x=12 does make the equation true.

• No, x=12 does not make the equation true.

Problem 4

○ Lisa solved the equation 𝑥 + 6 = 8 + 7𝑥 and claimed that

the solution is 𝑥 = − 1 3 .

○ Is she correct?

○ Explain.

Multiple Choice

Given the equation 𝑥 + 6 = 8 + 7𝑥 is the solution 𝑥 = − 1 3 . 

• Yes, the solution is that x=-13

• No, the solution is not that x=-13

Problem 5

○ Angel transformed the following equation from

6𝑥 + 4 − 𝑥 = 2(𝑥 + 1) to 10 = 2(𝑥 + 1).

○ He then stated that the solution to the equation is

𝑥 = 4.

○ Is he correct?

○ Explain.

Multiple Choice

The equation 6𝑥 + 4 − 𝑥 = 2(𝑥 + 1) was transformed to 10 = 2(𝑥 + 1)

Is the solution x=4?

• Yes, the solution for the equation x=4

• No, the solution for the equation is not x=4

• No, the solution is not x=4 because the solution is x=-4

• No, the solution is not x=4 because the original equation was not transformed correctly.

Problem 6

○ Claire was able to verify that 𝑥 = 3 was a solution to her

teacher’s linear equation, but the equation got erased

from the board.

○ What might the equation have been?

○ Identify as many equations as you can with a solution

of 𝑥 = 3

Open Ended

The solution is x = 3

What might the equation have been?

Problem 7

○ Does an equation always have a solution?

○ Could you come up with an equation that does not

have a solution?

Open Ended

Does an equation always have a solution?

Open Ended

Could you come up with an equation that does not have a solution?