1) The document discusses using the distributive property to solve linear equations and discusses how not all linear equations have solutions. It provides examples of using the distributive property to both simplify and expand expressions. 2) The distributive property is applied only to terms within a group and not to terms outside of the group. An equation has no solution if it can be transformed into an untrue statement when solving. 3) The document concludes by reminding students how to use the distributive property to transform equations and that some equations have no solutions. It provides recommendations for additional practice on IXL.

2. • Help Session – Link
• IXL –
Housekeeping

3. • Students transform equations into simpler
forms using the distributive property.
• Students learn that not every linear
equation has a solution.
Unit 2: Lesson 6
Solutions of a Linear Equation

4. Solve the linear equation
1
5
x + 13 + x = 1 − 9x + 22

5. The distributive property can be used to both expand and
simplify expressions.
We have already used the distributive property to “collect
like terms.”
For example, 2x + 6x = (2 + 6)x = 8x.
We have also used the distributive property to expand
expressions. For example, 2(x + 5) = 2x + 10.
The Distributive
Property

6. In this lesson, we will continue to use the distributive
property to solve more complicated equations.
Also highlighted in this lesson is a
common error that is made when
using the distributive property, which is
multiplying a factor to terms that are
not part of the group.
For example, in the expression 3(x + 1) − 5, students should
know that they do not distribute the factor 3 to the term −5
because it is not in the group (x + 1).

7. What value of x would make the linear equation 4x + 3(4x + 7) = 4(7x
+ 3) − 3 true? What is the “best” first step and why?
Example 1

8. What value of x would make the following linear equation true:
20 − (3x − 9) − 2 = −(−11x + 1)?
Example 2

9. What value of x would make the following linear equation true:
1
2
(4x + 6) − 2 = −(5x + 9)?
Example 3

10. Consider the following equation: 2(x + 1) = 2x − 3. What value of x
makes the equation true?
Example 4

11. Can you think of a value of x that would make this equation true?
2(x + 1) = 2x − 3
Why do you think this happened?
We know that an equation is a statement of equality. The linear
expressions were such that they could not be equal to each other, no
matter what value was substituted for x.
What can we conclude?

12. What value of x would make the following linear equation true:
9(4 − 2x) − 3 = 4 − 6(3x − 5)?
Try this one!

13. Can you think of an equation that has no solution???
Think of your own!

14. Find the value of 𝒙𝒙 that makes the equation true.
1. 17 − 5(2x − 9) = −(−6x + 10) + 4
2. −(x − 7) +
5
3
= 2(x + 9)
3.
4
9
+ 4(x − 1) =
28
9
− (x − 7x) + 1
4. 5(3x + 4) − 2x = 7x − 3(−2x + 11)
5. 7x − (3x + 5) − 8 =
1
2
(8x + 20) − 7x + 5
6. Write three equations that have no solution
Practice

15. The distributive property is used to expand expressions. For example,
the expression 2(3x − 10) is rewritten as 6x − 10 after the distributive
property is applied.
The distributive property is used to simplify expressions. For example,
the expression 7x + 11x is rewritten as (7 + 11)x and 18x after the
distributive property is applied.
The distributive property is applied only to terms within a group:
4(3x + 5) − 2 = 12x + 20 − 2.
Notice that the term -2 is not part of the group and, therefore, not
multiplied by 4.
When an equation is transformed into an untrue sentence, such as
5 ≠ 11, we say the equation has no solution.
Summary

16. Wrap Up
• We know how to transform equations into simpler forms
using the distributive property.
• We now know that there are some equations that do not
have solutions.
What now???
• IXL Skill U.7 – Solve Multi-Step Equations
• IXL Skill U.8 – Solve Equations Involving Like Terms
• IXL Skill U.9 – Solve Equations With Variables on Both Sides
• IXL Skill U.10 – Solve Equations: Mixed Review