1. 2.5 Solve Equations with variables on Both Sides
Essential Question: How do you solve multi-step equations?
Warm-up:
Solve the equation.
2m – 6 + 4m = 12
6a – 5 (a – 1) = 11.
3. A charter bus company charges $11.25 per ticket plus a handling charge of $.50 per ticket, and a $15 fee for booking the bus. If a group pays $297 a charter a bus, how many tickets did they buy?
Common
Core
CC.9-12.A.REI.3
Solve linear
equations and
inequalities in
one variable,
including
equations with
coefficients
represented by
letters.

2. Solve an equation with variables on both sides
EXAMPLE 1
Solve an equation with variables on both sides
Solve 7 – 8x = 4x – 17.
7 – 8x = 4x – 17
Write original equation.
7 – 8x + 8x = 4x – x
Add 8x to each side.
7 = 12x – 17
Simplify each side.
24 = 12x
Add 17 to each side.
2 = x
Divide each side by 12.
ANSWER
The solution is 2. Check by substituting 2 for x in the original equation.

3. Solve an equation with variables on both sides
EXAMPLE 1
Solve an equation with variables on both sides
CHECK
7 – 8x = 4x – 17
Write original equation.
7 – 8(2) = 4(2) – 17 ?
Substitute 2 for x.
–9 = 4(2) – 17 ?
Simplify left side.
–9 = –9
Simplify right side. Solution checks.

4. Solve an equation with grouping symbols
EXAMPLE 2
Solve an equation with grouping symbols
9x – 5 = 1 4
(16x + 60).
Solve 1 4
(16x + 60)
9x – 5 =
Write original equation.
9x – 5 = 4x + 15
Distributive property
5x – 5 = 15
Subtract 4x from each side.
5x = 20
Add 5 to each side.
x = 4
Divide each side by 5.

5. GUIDED PRACTICE
for Examples 1 and 2
Solve the equation. Check your solution.
– 3m = 5m 3
ANSWER

6. GUIDED PRACTICE
for Examples 1 and 2
Solve the equation. Check your solution.
c = 4c – 7
ANSWER 9

7. GUIDED PRACTICE
for Examples 1 and 2
Solve the equation. Check your solution.
– 3k = 17k – 2k
ANSWER –8

8. GUIDED PRACTICE
for Examples 1 and 2
Solve the equation. Check your solution.
z – 2 = 2(3z – 4)
ANSWER 6

9. GUIDED PRACTICE
for Examples 1 and 2
Solve the equation. Check your solution.
– 4a = 5(a – 3)
ANSWER 2

10. GUIDED PRACTICE
for Examples 1 and 2
Solve the equation. Check your solution.
8y – 6 = 2 3
(6y + 15) 6.
ANSWER 4

11. EXAMPLE 3
Solve a real-world problem
CAR SALES
A car dealership sold 78 new cars and 67 used cars this year. The number of new cars sold by the dealership has been increasing by 6 cars each year. The number of used cars sold by the dealership has been decreasing by 4 cars each year. If these trends continue, in how many years will the number of new cars sold be twice the number of used cars sold?

12. EXAMPLE 3
Solve a real-world problem
SOLUTION
Let x represent the number of years from now. So, 6x represents the increase in the number of new cars sold over x years and –4x represents the decrease in the number of used cars sold over x years. Write a verbal model. 67 78 + 6x
= 2 (
(– 4 x) )

13. Solve a real-world problem
EXAMPLE 3
Solve a real-world problem
78 + 6x = 2(67 – 4x)
Write equation.
78 + 6x = 134 – 8x
Distributive property
x = 134
Add 8x to each side.
14x = 56
Subtract 78 from each side.
x = 4
Divide each side by 14.
ANSWER
The number of new cars sold will be twice the number of used cars sold in 4 years.

14. EXAMPLE 3
Solve a real-world problem
CHECK
You can use a table to check your answer.
YEAR 1 2 3 4
Used car sold 67 63 59 55 51
New car sold 78 84 90 96
102

15. GUIDED PRACTICE
for Example 3 7.
WHAT IF? In Example 3, suppose the car dealership sold 50 new cars this year instead of 78. In how many years will the number of new cars sold be twice the number of used cars sold?
ANSWER
6 yr

16. Identify the number of solutions of an equation
EXAMPLE 4
Identify the number of solutions of an equation
Solve the equation, if possible.
a. 3x = 3(x + 4)
b. 2x + 10 = 2(x + 5)
SOLUTION
a. 3x = 3(x + 4)
Original equation
3x = 3x + 12
Distributive property
The equation 3x = 3x + 12 is not true because the number 3x cannot be equal to 12 more than itself. So, the equation has no solution. This can be demonstrated by continuing to solve the equation.

17. Identify the number of solutions of an equation
EXAMPLE 4
Identify the number of solutions of an equation
3x – 3x = 3x + 12 – 3x
Subtract 3x from each side.
0 = 12
Simplify.
ANSWER
The statement 0 = 12 is not true, so the equation has
no solution.

18. Identify the number of solutions of an equation
EXAMPLE 1
EXAMPLE 4
Identify the number of solutions of an equation
b. 2x + 10 = 2(x + 5)
Original equation
2x + 10 = 2x + 10
Distributive property
ANSWER
Notice that the statement 2x + 10 = 2x + 10 is true for all values of x. So, the equation is an identity, and the solution is all real numbers.

19. GUIDED PRACTICE
for Example 4
Solve the equation, if possible.
z + 12 = 9(z + 3)
ANSWER
no solution

20. GUIDED PRACTICE
for Example 4
Solve the equation, if possible.
w + 1 = 8w + 1
ANSWER

21. GUIDED PRACTICE
for Example 4
Solve the equation, if possible.
(2a + 2) = 2(3a + 3)
ANSWER
identity

22. Summary: Steps for Solving Linear Equations
Step 1: Use the distributive property to remove any grouping symbols. Step 2: Simplify the expression on each side of the equation. Step 3: Use properties of equality to collect the variable terms on one side of the equation and the constant terms on the other side of the equation. Step 4: Use properties of equality to solve for the variable. Step 5: Check your solution in the original equation.

23. Classwork/Homework
2.5 Exercises
2-54 even
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