Multistep Equations
Equations with Variables on Both Sides
Solving Equations with Fractions
Special Cases
Real-World Problems
100
Solve for p.
2(4p + 2) = 12
p = 1
100
Solve for x.
6x + 7 = 13 + 7x
r = -6
100
What could you multiply everything in the equation by to clear the fractions in this equation?
Any common multiple of 2, 8, 16.
100
How many solutions does the equation have?
2(x + 1) = 6x - 8
2x + 2 = 6x - 8
2 = 4x - 8
10 = 4x
10/4 = x
100
Derrick’s Dog Sitting and Darlene’s Dog Sitting are competing for new business. Derrick charges a one-time fee of $12.00 and an additional $5.00 per hour. Darlene charges a one-time fee of $18.00 and an additional $3.00 per hour. Write and solve an equation to find the number of hours for which the total cost will be the same for the two services.
12 + 5x = 18 + 3x
x = 3 hours
200
Solve for r.
r + 11 + 8r = 29
r = 2
200
-8 - x = -3x
x = 4
200
What multiple could you use to clear the fractions?
Any common multiple of 3 and 4 would work.
200
2(x − 3)+ 9 = 5 + x −1
one solution
200
Maria pays a yearly fee of $3 to her swimming club and $2 per lesson. Carmen pays a yearly fee of $5 and $2 per lesson. After how many lessons will Maria and Carmen have paid the same amount. Write an equation and explain your answer.
There is no solution. Carmen pays a higher yearly fee and pays the same per lesson. Therefore, Carmen's fees will always be higher.
300
Solve for y.
5y - 3(2 - y) = 10
y = 2
300
Solve for x.
-7x - 3x + 2 = -8x - 8
x = 5
300
Solve the equation.
x = 1
300
2(x +10)−17 = 5 + 2x − 2
infinitely many solutions
300
Andy’s Rental Car charges an initial fee of $20 plus an additional $30 per day to rent a car. Buddy’s Rental Car charges an initial fee of $36 plus an additional $28 per day. For what number of days is the total cost charged the same for both companies?
20 + 30h = 36 + 28h
h = 8 hours
400
Solve for m.
-13 = 5(1 + 4m) - 2m
m = -1
400
4n - 40 = 7(-2n + 2)
n = 3
400
Solve the equation.
m = 8
400
5x + 3− 3x = 2(x + 3)− 5
no solution
400
A water tank holds 256 gallons but is leaking at a rate of 3 gallons per week. A second water tank holds 384 gallons but is leaking at a rate of 5 gallons per week. After how many weeks will the amount of water in the two tanks be the same?
256 - 3w = 384 - 5w
w = 64 weeks
500
If the perimeter of the rectangle is 96 feet, what are the length and width of the rectangular section?
Length: 39 feet
Width: 9 feet
500
Solve for m.
8m + 4(5m - 3) = 4(6m + 4) - 4
m = 6
500
Solve the equation.
x = 36/72
x = 1/2
x = 0.5
500
4(x + 3) - 4 = 8(1/2x + 1)
4x + 12 - 4 = 4x + 8
4x +8 = 4x + 8
infinitely many solutions
500
At Silver Gym, membership is $25 per month, and personal training sessions are $30 each. At Fit Factor, membership is $65 per month, and personal training sessions are $20 each. In one month, how many personal training sessions would Sarah have to buy to make the total cost at the two gyms equal?
25 + 30s = 65 + 20s
s = 4 sessions