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3.4
Solve Equations with Variables on Both Sides
Warm Up
Lesson Presentation
Lesson Quiz
3.4
Warm-Up
Solve the equation.
1. 2m – 6 + 4m = 12
ANSWER
3
2. 6a – 5(a – 1) = 11
ANSWER
6
3.4
Warm-Up
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 to
charter a bus, how many tickets did they buy?
ANSWER
24 tickets
3.4
Example 1
Solve 7 – 8x = 4x – 17.
7 – 8x = 4x – 17
7 – 8x + 8x = 4x – 17 + 8x
7 = 12x – 17
24 = 12x
2=x
Write original equation.
Add 8x to each side.
Simplify each side.
Add 17 to each side.
Divide each side by 12.
ANSWER
The solution is 2. Check by substituting 2 for x in the
original equation.
3.4
Example 1
CHECK
7 – 8x = 4x – 17
?
7 – 8(2) = 4(2) – 17
?
Write original equation.
Substitute 2 for x.
–9 = 4(2) – 17
Simplify left side.
–9 = –9
Simplify right side. Solution checks.
3.4
Guided Practice
Solve the equation. Check your solution.
1.
24 – 3m = 5m
ANSWER
2.
20 + c = 4c – 7
ANSWER
3.
3
9
9 – 3k = 17k – 2k
ANSWER
–8
3.4
Example 2
1
Solve 9x – 5 = 4 (16x + 60).
1
9x – 5 = (16x + 60)
4
Write original equation.
9x – 5 = 4x + 15
Distributive property
5x – 5 = 15
Subtract 4x from each side.
5x = 20
x=4
Add 5 to each side.
Divide each side by 5.
3.4
Guided Practice
Solve the equation. Check your solution.
4. 5z – 2 = 2(3z – 4)
ANSWER
6
5. 3 – 4a = 5(a – 3)
ANSWER
6.
2
2
8y – 6 = 3 (6y + 15)
ANSWER
4
3.4
Example 3
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?
3.4
Example 3
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.
78
+
6x
=2(
67
+
(– 4 x)
)
3.4
Example 3
78 + 6x = 2(67 – 4x)
Write equation.
78 + 6x = 134 – 8x
Distributive property
78 + 14x = 134
14x = 56
x= 4
Add 8x to each side.
Subtract 78 from each side.
Divide each side by 14.
ANSWER
The number of new cars sold will be twice the
number of used cars sold in 4 years.
3.4
Example 3
CHECK
You can use a table to check your answer.
3.4
7.
Guided Practice
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
3.4
Example 4
Solve the equation, if possible.
a. 3x = 3(x + 4)
b. 2x + 10 = 2(x + 5)
SOLUTION
a. 3x = 3(x + 4)
3x = 3x + 12
Original equation
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.
3.4
Example 4
3x – 3x = 3x + 12 – 3x
0 = 12
Subtract 3x from each side.
Simplify.
ANSWER
The statement 0 = 12 is not true, so the equation has
no solution.
3.4
b.
Example 4
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.
3.4
Guided Practice
Solve the equation, if possible.
8.
9z + 12 = 9(z + 3)
ANSWER
no solution
9. 7w + 1 = 8w + 1
ANSWER
0
10. 3(2a + 2) = 2(3a + 3)
ANSWER
identity
3.4
Lesson Quiz
Solve the equation, if possible.
1. 3(3x + 6) = 9(x + 2)
ANSWER
2.
The equation is an identity.
7(h – 4) = 2h + 17
ANSWER
9
3. 8 – 2w = 6w – 8
ANSWER
2
4.
4g + 3 = 2(2g + 3)
ANSWER
The equation has no solution.
3.4
5.
Lesson Quiz
Bryson is looking for a repair service for
general household maintenance. One service
charges $75 to join the service and $30 per
hour. Another service charge $45 per hour.
After how many hours of service is the total
cost for the two services the same?
ANSWER
5h