Linear Equation WPs
Samantha is a prom photographer. She charges a flat fee to book a photoshoot and an hourly fee during the photoshoot. The following equation represents her total price: y = 15x + 25
What does the 15 represent in the equation?
A) The total number of hours the photoshoot lasts
B) The flat fee charged to book a photoshoot
C) The total price charged for the photoshoot, excluding the flat fee
D) The hourly fee charged during the photoshoot
D) The hourly fee charged during the photoshoot.
Explanation: In the equation, y = 15x + 25, the term 15x represents the total cost for a photoshoot that lasts x hours (excluding the flat fee). However, the 15 only represents the hourly fee. The constant term, 25, represents the flat fee charged to book the photoshoot, which is added to the total price.
A) 5
B) 6
C) 7
D) 8
Correct Answer: B) 6
Explanation: Subtract 5 from both sides to isolate 2x. Then, divide by 2 to solve for x:
2x+5=15
2x=15-5
2x=10
x=5
Solve for x:
3x + 7 < 22
Correct answer: x < 5
Explanation: Subtract 7 from both sides. Divide both sides by 3.
3x + 7 < 22
3x < 15
x < 5
Solve for y.
w = v – x + y
(Hint: Isolate the variable y by performing inverse operations)
Correct answer: y = w - v + x
At Tom's toy store, the total daily expenses are the cost of rent and the cost of purchasing new toys. The rent for the store is $500, and on average, each new toy costs $10. This expression shows the total daily expenses for Tom's toy store to buy x new toys: 500 + 10x. What does the term 500 represent?
A) The total cost of new toys for the day
B) The number of new toys Tom buys in a day
C) The total daily expenses for Tom's toy store
D) The cost of rent for the store
D) The cost of rent for the store.
Explanation: In the expression 500 + 10x, the term 500 represents the initial cost of rent for Tom's toy store. The term 10x represents the cost of purchasing each new toy, where x is the number of new toys purchased. Adding these two terms gives the total daily expenses for Tom's toy store.
Correct Answer: D) 9
Explanation: Distribute the 4 on the left side. Subtract 2x from both sides. Add 12 to both sides. Dive 2 on both sides.
4(x-3)=2x+6
4x-12=2x+6
2x-12=6
2x=18
x=9
Solve for x:
-3x + 7 < 22
Correct answer: x > -5
Explanation: Subtract 7 from both sides. Divide both sides by -3. Recall dividing by a negative number causes the direction of the inequality symbol to flip.
-3x + 7 < 22
-3x < 15
x > -5
The formula used to find the volume of a cylinder is
V = πr2h
Rearrange the formula to find the height, h, of a cylinder.
(Hint: Isolate the variable h)
Correct answer: h = V/πr2
Jessica sells handmade scarves at a local market. She uses the equation P=10s−(3s+20) to calculate her total profit, in dollars, when she sells s scarves. How much does she charge for each scarf?
A) $3.00
B) $7.00
C) $10.00
D) $20.00
C) $10.00
Explanation: To find how much Jessica charges per scarf, look at the equation: P=10s−(3s+20). The term 10s represents the revenue from selling s scarves at $10.00 each. Therefore, the correct answer is C) $10.00.
The -$3.00 term in the equation represents the cost of making each scarf, not the selling price.
Solve for x:
5x + 9 = 3(x + 7)
A) -9
B) 12
C) -3
D) 6
Correct Answer: D) 6
Distribute the 3 on the right side. Subtract 3x from both sides. Subtract 9 from both sides. Divide 2 on both sides.
5x+9=3(x+7)
5x+9=3x+21
2x+9=21
2x=12
x=6
Solve for x:
3(2x−4)+5 ≥ 7x−3
Correct answer: x≤−4.
The solution set is x≤−4.
Solve for q in terms of r and s.
r = sq
Correct answer: q = r/s
Sarah sells treats at a local bake sell. She uses the equation P=8t−(2t+20) to calculate her total profit, in dollars, when she sells t treats. If Sarah wants to make a profit of $40, how many treats does she need to sell?
A) 5
B) 10
C) 15
D) 20
Correct answer: B
Explanation:
In the equation, P=8t−(2t+20), the term 8t represents the money made from selling t treats at $8 each, and the term -(2t+20) represents the total cost to make the treats. To find the number of treats Sarah needs to sell to make a profit of $60, we set P to $40:
40=8t−(2t+20)
Simplifying, we get:
40=8t−2t−20
40=6t−20
Adding 20 to both sides:
60=6t
Dividing by 6:
60/6=t
10=t
So, Sarah needs to sell 10 treats to make a profit of $40.
Solve for x:
2(3x + 5) = 5(2x - 1)
Correct answer: x=15/4 or 3.75
Explanation: Distribute the 2 on the left side. Distribute the 5 on the right side. Subtract 10x on both sides. Subtract 10 on both sides. Divide 4 on both sides. Simplify the fraction.
2(3x+5)=5(2x-1)
6x+10=10x-5
-4x+10=-5
-4x=-15
x=15/4 or 3.75
The local gym charges a monthly membership fee plus a fee for each personal training session. The total cost, C, can be represented by the inequality C ≥ 80 + 30n, where n is the number of training sessions. If Jane can afford up to $200 per month for the gym, what is the maximum number of training sessions she can attend?
A) 4 sessions
B) 5 sessions
C) 6 sessions
D) 7 sessions
Correct answer: A) 4 sessions
Explanation: Substitute $200 for C in the inequality. Subtract 80 from both sides. Divide 30 on both sides.
$200 ≥ 80 + 3
$120 ≥ 30n
4 ≥ n
Jane can attend a maximum of 4 training sessions.
Solve for f in terms of T.
T=1/f
Correct answer: f = 1/T
Amanda is planning a fundraiser for her school's music program. She wants to sell candy bars and cookies to raise money. Each candy bar costs $1.25, and each cookie costs $0.75. Amanda plans to sell a total of 200 items. If she wants to raise $200, how many candy bars and cookies should she sell?
100 candy bars
100 cookies
Explanation: To solve this problem, students would need to set up a system of equations. Let x represent the number of candy bars and y represent the number of cookies. The total number of items sold is x+y=200, and the total amount raised is 1.25x+0.75y=200. Students would need to find the solution to the system to find the values of x and y, representing the number of candy bars and cookies Amanda should sell.
Solve for x:
3(x - 4) + 2 = 3x-4(3x + 1)
Correct answer: x=12/6 or 1/2 or 0.5
Explanation: Distribute 3 on the left side. Distribute -4 on the right side. Combine like terms (-12+2) on the left side. Combine like terms (3x-12x) on the right side. Add 9x on both sides. Add 10 on both sides. Divide 12 on both sides.
3(x-4)+2=3x-4(3x+1)
3x-12+2=3x-4(3x+1)
3x-12+2=3x-12x-4
3x-10=3x-12x-4
3x-10=-9x-4
12x-10=-4
12x=6
x=12/6 or 1/2 or 0.5
A catering company charges a fee for renting its banquet hall plus a fee per person for food. The total cost, C, for a party with x guests can be represented by the inequality C ≥ 500 + 30x. If the maximum budget for the party is $1000, what is the maximum number of guests allowed?
Correct answer: 16 ≥ x (x is less than or equal to 16)
Explanation: Substitute $1000 for C in the inequality. Subtract 500 from both sides. Divide by 30 on both sides.
C ≥ 500 + 30x
1000 ≥ 500 + 30x
500 ≥ 30x
16.67 ≥ x
The context of this problem is important. You cannot have 0.67 of a person, and rounding up to 17 would make the company go over budget.
Solve for q in terms of p, r, and s.
p=(1/r)qs