Transcript la1_ch02_05 solving equations with variables on

2.5 Solving equations with
variables on both sides
• You will solve
equations with variables
on both sides.
• Essential Question:
How do you solve
equations with variables
on both sides?
Warm-Up1Exercises
EXAMPLE
Solve an equation with variables on both sides
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.
Warm-Up1Exercises
EXAMPLE
Solve an equation with variables on both sides
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.
Warm-Up2Exercises
EXAMPLE
Solve an equation with grouping symbols
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.
Warm-Up
Exercises
GUIDED
PRACTICE
for Examples 1 and 2
Solve the equation. Check your solution.
1. 24 – 3m = 5m
ANSWER
3
Warm-Up
Exercises
GUIDED
PRACTICE
for Examples 1 and 2
Solve the equation. Check your solution.
2. 20 + c = 4c – 7
ANSWER
9
Warm-Up
Exercises
GUIDED
PRACTICE
for Examples 1 and 2
Solve the equation. Check your solution.
3. 9 – 3k = 17k – 2k
ANSWER
–8
Warm-Up
Exercises
GUIDED
PRACTICE
for Examples 1 and 2
Solve the equation. Check your solution.
4. 5z – 2 = 2(3z – 4)
ANSWER
6
Warm-Up
Exercises
GUIDED
PRACTICE
for Examples 1 and 2
Solve the equation. Check your solution.
5. 3 – 4a = 5(a – 3)
ANSWER
2
Warm-Up
Exercises
GUIDED
PRACTICE
for Examples 1 and 2
Solve the equation. Check your solution.
6.
2
8y – 6 = 3 (6y + 15)
ANSWER
4
EXAMPLE
Warm-Up3Exercises
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?
EXAMPLE
Warm-Up3Exercises
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.
78
+
6x
=2(
67
+
(– 4 x)
)
EXAMPLE
Warm-Up3Exercises
Solve a real-world problem
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.
EXAMPLE
Warm-Up3Exercises
Solve a real-world problem
CHECK
You can use a table to check your answer.
YEAR
Used car sold
0
67
1
63
2
59
3
55
4
51
New car sold
78
84
90
96
102
Warm-Up
Exercises
GUIDED
PRACTICE
7.
for Example 3
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
Warm-Up4Exercises
EXAMPLE
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.
Warm-Up4Exercises
EXAMPLE
Identify the number of solutions of an equation
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.
Warm-Up4
EXAMPLE
1Exercises
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.
Warm-Up
Exercises
GUIDED
PRACTICE
for Example 4
Solve the equation, if possible.
8.
9z + 12 = 9(z + 3)
ANSWER
no solution
Warm-Up
Exercises
GUIDED
PRACTICE
for Example 4
Solve the equation, if possible.
9. 7w + 1 = 8w + 1
ANSWER
0
Warm-Up
Exercises
GUIDED
PRACTICE
for Example 4
Solve the equation, if possible.
10. 3(2a + 2) = 2(3a + 3)
ANSWER
identity
Daily
Homework
Quiz
Warm-Up
Exercises
Solve the equation, if possible.
1.
3(3x + 6) = 9(x + 2)
ANSWER
2.
7(h – 4) = 2h + 17
ANSWER
3.
The equation is an identity.
9
8 – 2w = 6w – 8
ANSWER
2
Daily
Homework
Quiz
Warm-Up
Exercises
4.
4g + 3 = 2(2g + 3)
ANSWER
5.
The equation has no solution.
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
• You will solve
equations with variables
on both sides.
•To solve equations with
variables on both sides,
collect the variable terms on
one side and the constant
terms on the other.
• Some equations, called
identities, are true for all
values of the variable. Other
equations have no solutions.
• Essential Question:
How do you solve
equations with variables
on both sides?
To solve equations with variables
on both sides, first simplify the
expressions on each side of the
equation by using the distributive
property to remove grouping
symbols
and then combining like terms.
Next, use properties of equality to
collect variable terms on one side
of the equation and constants on
the other. Then solve the equation
by isolating the variable.