Multi-Step Equations

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Multi-Step Equations · Grade 7 · Worksheet 3

1. Noah is saving up to buy a new bicycle that costs $316. He already has $76 saved from his birthday money. For the next 6 weeks, he plans to save the same amount each week from his allowance. How much money does Noah need to save each week to have exactly enough for the bicycle? Answer: $40 Solution: Let w represent the amount Noah saves each week. Noah already has $76, and he will save for 6 weeks, so his total savings will be 76 + 6w. This total must equal the cost of the bicycle, $316.
   Full step-by-step solution

   Step 1: Let w represent the amount Noah saves each week. Step 2: Noah already has $76, and he will save for 6 weeks, so his total savings will be 76 + 6w. Step 3: This total must equal the cost of the bicycle, $316. Step 4: Write the equation: 76 + 6w = 316. Step 5: Subtract 76 from both sides: 6w = 316 - 76 = 240. Step 6: Divide both sides by 6: w = 240 / 6 = 40. Step 7: Noah needs to save $40 each week. Final answer: $40.

2. Mere is saving money to buy a new laptop that costs $1,024. She already has $256 saved. For each of the next 8 weeks, she plans to save the same amount from her part-time job. After those 8 weeks, she will also receive a $48 birthday gift from her grandmother. How much does Mere need to save each week to have exactly enough for the laptop? Answer: 90 Solution: Let x = amount saved each week.
   Full step-by-step solution

   Step 1: Let x = amount saved each week. Step 2: Total saved = starting amount + (weekly amount × 8 weeks) + birthday gift Step 3: Equation: 256 + 8x + 48 = 1024 Step 4: Combine like terms: 256 + 48 = 304, so 304 + 8x = 1024 Step 5: Subtract 304 from both sides: 8x = 1024 - 304 = 720 Step 6: Divide both sides by 8: x = 720 ÷ 8 = 90 Step 7: Mere needs to save $90 each week. The answer is 90.

3. 3(2x - 7) + 5 = 4(x + 2) - 3 Answer: x = 9 Solution: 3(2x - 7) + 5 = 4(x + 2) - 3 6x - 21 + 5 = 4x + 8 - 3 6x - 16 = 4x + 5 Get all x terms on one side by subtracting 4x from both sides 6x - 16 - 4x = 4x + 5 - 4x 2x - 16 = 5 Isolate the x term by adding 16 to both sides 2x - 16 + 16 = 5 + 16 2x = 21 Solve for x by dividing both sides by 2 2x ÷ 2 =…
   Full step-by-step solution

   Step 1: Distribute on both sides 3(2x - 7) + 5 = 4(x + 2) - 3 6x - 21 + 5 = 4x + 8 - 3 Step 2: Combine like terms on each side 6x - 16 = 4x + 5 Step 3: Get all x terms on one side by subtracting 4x from both sides 6x - 16 - 4x = 4x + 5 - 4x 2x - 16 = 5 Step 4: Isolate the x term by adding 16 to both sides 2x - 16 + 16 = 5 + 16 2x = 21 Step 5: Solve for x by dividing both sides by 2 2x ÷ 2 = 21 ÷ 2 x = 10.5 The answer is x = 10.5.

4. Mason is saving money to buy a new gaming console that costs $427. He already has $127 saved from his birthday. He plans to mow lawns in his neighborhood to earn the rest. If Mason charges $22 per lawn, how many lawns does he need to mow to have exactly enough money for the console? Answer: x = 14 Solution: Let x represent the number of lawns Mason needs to mow. Write an equation: money saved + money from lawns = total cost 127 + 22x = 427 Subtract 127 from both sides to isolate the term with x: 127 + 22x - 127 = 427 - 127 22x = 300 Divide both sides by 22 to solve for x: 22x / 22 = 300 / 22 x =…
   Full step-by-step solution

   Step 1: Let x represent the number of lawns Mason needs to mow. Step 2: Write an equation: money saved + money from lawns = total cost 127 + 22x = 427 Step 3: Subtract 127 from both sides to isolate the term with x: 127 + 22x - 127 = 427 - 127 22x = 300 Step 4: Divide both sides by 22 to solve for x: 22x / 22 = 300 / 22 x = 13.636... Since Mason cannot mow a fraction of a lawn, we round up to the next whole number. Step 5: Check: 127 + 22(14) = 127 + 308 = 435. This is $8 more than $427, but if he mows 13 lawns, he would have 127 + 286 = 413, which is not enough. So he must mow 14 lawns. Mason needs to mow 14 lawns to have enough money for the console.

5. Liam is saving money to buy a new video game that costs $65. He already has $28 saved from his allowance. For the next 4 weeks, he plans to save the same amount each week from his part-time job. How much money does Liam need to save each week to have exactly enough for the video game after 4 weeks? Answer: 9.25 Solution: Determine how much more money Liam needs. The game costs $65. He already has $28 saved.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Determine how much more money Liam needs.** The game costs $65. He already has $28 saved. Money still needed = 65 - 28 65 - 28 = 37 So, Liam needs $37 more. --- **Step 2: Understand the saving plan.** He will save the same amount each week for 4 weeks. Let the amount saved per week = \( x \). In 4 weeks, he will save \( 4 \times x \). --- **Step 3: Set up the equation.** The money saved in 4 weeks must equal $37. So: 4x = 37 --- **Step 4: Solve for x.** Divide both sides by 4: x = 37 / 4 37 ÷ 4 = 9.25 --- **Step 5: Interpret the result.** Liam needs to save $9.25 each week for 4 weeks to reach his goal. --- **Final answer:** 9.25

6. A school is organizing a field trip and needs to rent buses. Each bus can hold 48 students. There are 312 students going on the trip. The school also needs to reserve 2 buses for teachers and chaperones. If each bus costs $120 to rent, what is the total cost for all the buses needed? Answer: 1080 Solution: Calculate how many buses are needed for the 312 students 312 students ÷ 48 students per bus = 6.5 buses Since we can't have half a bus, we need to round up to 7 buses for students 7 buses for students + 2 buses for teachers/chaperones = 9 total buses 9 buses × $120 per bus = $1080 The total cost…
   Full step-by-step solution

   Step 1: Calculate how many buses are needed for the 312 students 312 students ÷ 48 students per bus = 6.5 buses Since we can't have half a bus, we need to round up to 7 buses for students Step 2: Add the buses for teachers and chaperones 7 buses for students + 2 buses for teachers/chaperones = 9 total buses Step 3: Calculate the total cost 9 buses × $120 per bus = $1080 The total cost for all buses is $1080.

7. Emma is designing a rectangular mural for her school art project. The length of the mural is 3 meters more than twice its width. If the perimeter of the mural is 54 meters, what is the width of the mural? Answer: 8 Solution: Let w represent the width of the mural in meters. The length is 3 meters more than twice the width, so length = 2w + 3. The formula for the perimeter of a rectangle is P = 2(length + width).
   Full step-by-step solution

   Step 1: Let w represent the width of the mural in meters. Step 2: The length is 3 meters more than twice the width, so length = 2w + 3. Step 3: The formula for the perimeter of a rectangle is P = 2(length + width). Step 4: Substitute the known values: 54 = 2((2w + 3) + w) Step 5: Simplify inside the parentheses: 54 = 2(3w + 3) Step 6: Distribute the 2: 54 = 6w + 6 Step 7: Subtract 6 from both sides: 48 = 6w Step 8: Divide both sides by 6: w = 8 Step 9: The width of the mural is 8 meters.

8. 3(2x - 5) + 7 = 4x + 13 Answer: x = 10.5 Solution: 3(2x - 5) + 7 = 4x + 13 Distribute the 3 Multiply 3 by each term inside the parentheses: 3 * 2x = 6x 3 * (-5) = -15 6x - 15 + 7 = 4x + 13 -15 + 7 = -8 6x - 8 = 4x + 13 Subtract 4x from both sides: 6x - 4x - 8 = 4x - 4x + 13 2x - 8 = 13 Add 8 to both sides: 2x - 8 + 8 = 13 + 8 2x = 21 Divide both…
   Full step-by-step solution

   Let's solve the equation step-by-step. We start with: 3(2x - 5) + 7 = 4x + 13 --- **Step 1: Distribute the 3** Multiply 3 by each term inside the parentheses: 3 * 2x = 6x 3 * (-5) = -15 So the equation becomes: 6x - 15 + 7 = 4x + 13 --- **Step 2: Combine like terms on the left** -15 + 7 = -8 So now: 6x - 8 = 4x + 13 --- **Step 3: Move x terms to one side** Subtract 4x from both sides: 6x - 4x - 8 = 4x - 4x + 13 2x - 8 = 13 --- **Step 4: Move constant terms to the other side** Add 8 to both sides: 2x - 8 + 8 = 13 + 8 2x = 21 --- **Step 5: Solve for x** Divide both sides by 2: x = 21/2 x = 10.5 --- **Final answer:** x = 10.5