Solve It!
Distribute & Solve
How Many Solutions?
Set Them Equal
Equal with Parentheses
100
Solve: 8x − 13 = 35.
Answer: x = 6
100
Solve: 5(x − 3) = 40.
Answer: x = 11
100
How many solutions? Explain: 7x + 6 = 7x − 10.
Answer: No Solution because the statement when solved is false.
No solution: same coefficients, different constants when simplified.
100
A robot has traveled 14 meters and moves 3 meters each second. Another has traveled 4 meters and moves 5 meters each second.
Write an equation for when their total distances are equal. Let t be the additional seconds.
Do not solve.
Answer: 14 + 3t = 4 + 5t
100
Four identical snack boxes each contain x crackers and 6 pretzels. Three identical boxes each contain 2x crackers and 2 pretzels. The two groups contain the same total number of pieces. Write an equation. Do not solve.
Answer: 4(x + 6) = 3(2x + 2)
200
Solve: 9x + 4 = 5x − 24.
Answer: x = -7
200
Solve: 3(x + 7) = x + 37.
Answer: x = 8
200
How many solutions? Explain: 5(2x + 3) = 10x + 15.
Answer: Infinitely many solutions because the expressions when simplified are equivalent.
200
A candle is 30 cm tall and burns 2 cm each hour. Another is 24 cm tall and burns 1 cm each hour. After how many hours will their heights be equal?
Write an equation and solve.
Answer: 30 − 2h = 24 − h; h = 6 hours
200
Three identical shelves each hold x novels and 8 biographies. Four other shelves each hold x novels and 3 biographies. The two groups contain the same total number of books. How many novels are on each shelf? Write an equation and solve.
Answer: 3(x + 8) = 4(x + 3); x = 12
300
Solve: 6x + 7 − 2x = x + 34.
Answer: x = 9.
300
Solve: 4(2x − 5) + 3x = 57.
Answer: x = 7
300
How many solutions? Explain
4(3x − 2) = 9x + 7.
Answer: One solution because when solved, x = 5.
300
Two machines produce labels.
Machine A has already produced 28 labels and continues producing 6 labels per minute.
Machine B has already produced 76 labels and continues producing 2 labels per minute.
After how many minutes, m, will both machines produce the same total number of labels?
Write and equation and solve.
Answer: 28 + 6m = 76 + 2m; m = 12
300
A craft club prepares two groups of bead kits. Each bag contains the same number of blue beads.
The first group has 5 kits. Each kit contains 2 bags of blue beads and 3 loose white beads.
The second group has 3 kits. Each kit contains 3 bags of blue beads and 9 loose white beads.
The two groups contain the same total number of beads. How many blue beads are in each bag?
Write and solve an equation.
Answer: 5(2x + 3) = 3(3x + 9); x = 12.
400
Solve: 1.8x − 4.5 = 0.6x + 3.9.
Answer: x = 7
400
Solve: 7 − 2(3x + 4) = 5x + 21.
Answer: x = -2
400
What coefficient k makes 3(4x − 5) = kx + 9 have no solution? Explain.
Answer: k = 12 because when substituted the equation simplifies to a false statement.
No solution: same coefficients, different constants when simplified.
400
Tank A contains 96 liters of water and drains 3.5 liters per minute. Tank B contains 60 liters of water and drains 1.5 liters per minute.
When will they contain equal amounts?
Write an equation, then find that amount.
Answer: 96 − 3.5t = 60 − 1.5t; when t = 18 the amount of water is 33 liters.
400
A rectangular mural has a perimeter of 156 feet. Its length is 5 times its width.
What are the length and width of the mural?
Let w = the width.
Write an equation and show your work.
Answer:. 2(w + 5w) = 156 or 5w + w + 5w + w = 156 (or any equivalent equation)
Width:13ft. Length:65ft.
Perimeter: the sum of the sides of a shape.
500
Solve: 5/6x − 4 = 1/3x + 2.
Answer: x = 12
500
Solve and check: 2/3(6x − 9) = 3(x + 2)
Answer: x = 12
500
Find a and b so 2(ax − 4) + 3x = 13x + b has infinitely many solutions.
Answer a = 6 and b = −8. For the equation to have infinitely many solutions, the expressions have to be equivalent.
500
A game player starts with 125 points and loses 4 points each round. Another starts with 35 points and gains 6 points each round. Let r = # of rounds.
When their scores are equal, what is each score?
Write an equation and solve.
Answer: 125 − 4r = 35 + 6r; 90 = 10r; r = 9. Each score is 89 points.
500
A theater has 7 rows with x + 5 seats per row and 8 extra seats. Another theater has 5 rows with 2x − 1 seats per row and 12 extra seats. They have the same number of seats. How many seats does each theater have?
Write an equation and solve.
Answer: 7(x + 5) + 8 = 5(2x − 1) + 12; x =12; there are 127 in each theater.
