Create Systems

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Answer Key & Explanations

Create Systems · Grade 9 · Worksheet 3

1. Charlotte is selling two types of handmade candles. She sold 9 large candles and 13 small candles for a total of $287. A large candle costs $8 more than a small candle. Write a system of equations to represent this situation, using L for the price of a large candle and S for the price of a small candle. Answer: L = S + 8, 9L + 13S = 287 Solution: Define variables. Let L = price of a large candle, S = price of a small candle. A large candle costs $8 more than a small candle, so L = S + 8.
   Full step-by-step solution

   Step 1: Define variables. Let L = price of a large candle, S = price of a small candle. Step 2: A large candle costs $8 more than a small candle, so L = S + 8. This is the first equation. Step 3: She sold 9 large candles and 13 small candles for a total of $287. Revenue from large candles is 9L, from small candles is 13S. Total revenue: 9L + 13S = 287. This is the second equation. Step 4: The system of equations is: L = S + 8 9L + 13S = 287 The answer is L = S + 8, 9L + 13S = 287.

2. Noah is selling two types of handmade candles. He sold 9 large candles and 8 small candles for a total of $287. A large candle costs $7 more than a small candle. Write a system of equations to represent this situation, using L for the price of a large candle and S for the price of a small candle. Answer: L = S + 7, 9L + 8S = 287 Solution: Define variables. Let L = price of a large candle, S = price of a small candle. A large candle costs $7 more than a small candle, so L = S + 7.
   Full step-by-step solution

   Step 1: Define variables. Let L = price of a large candle, S = price of a small candle. Step 2: A large candle costs $7 more than a small candle, so L = S + 7. This is the first equation. Step 3: He sold 9 large candles and 8 small candles for a total of $287. Revenue from large candles is 9L, from small candles is 8S. Total revenue is 9L + 8S = 287. This is the second equation. Step 4: The system of equations is: L = S + 7 9L + 8S = 287 The answer is L = S + 7, 9L + 8S = 287.

3. Mason is analyzing a rectangular garden on a coordinate grid. The garden's length is 7 meters more than twice its width. The perimeter of the garden is 62 meters. Write a system of two equations to represent this situation, where L represents the length and W represents the width. Do not solve the system. Answer: L = 2W + 7 and 2L + 2W = 62 Solution: Let L represent the length in meters and W represent the width in meters. The problem states that the length is 7 meters more than twice the width. This translates to: L = 2W + 7.
   Full step-by-step solution

   Step 1: Let L represent the length in meters and W represent the width in meters. Step 2: The problem states that the length is 7 meters more than twice the width. This translates to: L = 2W + 7. Step 3: The perimeter of a rectangle is given by the formula P = 2L + 2W. The problem says the perimeter is 62 meters, so: 2L + 2W = 62. Step 4: The system of two equations is: L = 2W + 7 2L + 2W = 62 The answer is L = 2W + 7 and 2L + 2W = 62.

4. A technology startup is analyzing two different pricing models for their new software subscription service. Model X has a setup fee of $500 plus $25 per user per month, while Model Y has no setup fee but charges $40 per user per month. The company wants to determine at what number of users both models would generate the same total monthly revenue. Write a system of equations to represent this situation and solve for the number of users and the total revenue at that point. Answer: x = 33.33 users, y = $1333.33 Solution: When two different pricing or cost structures are compared, we can set up equations where one variable represents the quantity (like number of users) and another represents the total cost or revenue.
   Full step-by-step solution

   This type of problem involves creating a system of linear equations to model real-world scenarios. When two different pricing or cost structures are compared, we can set up equations where one variable represents the quantity (like number of users) and another represents the total cost or revenue. The point where both equations are equal represents the break-even point between the two models.

5. Charlotte is selling two types of handmade scarves. She sold 12 knit scarves and 7 woven scarves for a total of $287. A knit scarf costs $7 more than a woven scarf. Write a system of equations to represent this situation, using k for the price of a knit scarf and w for the price of a woven scarf. Answer: k = w + 7, 12k + 7w = 287 Solution: Define variables. Let k = price of a knit scarf, w = price of a woven scarf. A knit scarf costs $7 more than a woven scarf, so k = w + 7.
   Full step-by-step solution

   Step 1: Define variables. Let k = price of a knit scarf, w = price of a woven scarf. Step 2: A knit scarf costs $7 more than a woven scarf, so k = w + 7. This is the first equation. Step 3: She sold 12 knit scarves and 7 woven scarves for a total of $287. The revenue from knit scarves is 12k, and from woven scarves is 7w. The total revenue is 12k + 7w = 287. This is the second equation. Step 4: The system of equations is: k = w + 7 12k + 7w = 287 The answer is k = w + 7, 12k + 7w = 287.

6. Isabella is selling two types of handmade candles. The large candles sell for $12 each and the small candles sell for $8 each. In one day, she sold 25 candles and collected $260. Write a system of equations to represent this situation, using L for large candles and S for small candles. Answer: L + S = 25, 12L + 8S = 260 Solution: When creating systems of equations from word problems, identify what each variable represents and what relationships are given.
   Full step-by-step solution

   When creating systems of equations from word problems, identify what each variable represents and what relationships are given. The total number of items gives one equation, while the total value or cost gives another equation. Make sure both equations use the same variables consistently.

7. Matiu is designing a rectangular garden on a coordinate grid. The garden's perimeter is 42 units. The length of the garden is 4 units more than twice its width. Write a system of two equations that represents this situation, using L for length and W for width. Do not solve the system. Answer: 2L + 2W = 42 and L = 2W + 4 Solution: Define the variables. Let L represent the length of the garden and W represent the width of the garden. The perimeter of a rectangle is calculated as 2 times the length plus 2 times the width.
   Full step-by-step solution

   Step 1: Define the variables. Let L represent the length of the garden and W represent the width of the garden. Step 2: Write the equation for the perimeter. The perimeter of a rectangle is calculated as 2 times the length plus 2 times the width. The problem states the perimeter is 42 units. So, the first equation is: 2L + 2W = 42. Step 3: Write the equation for the relationship between length and width. The problem states 'the length is 4 units more than twice its width'. 'Twice its width' means 2W. '4 units more than' means add 4. So, the length (L) equals 2W + 4. The second equation is: L = 2W + 4. Step 4: Combine both equations to form the system. System of equations: 2L + 2W = 42 L = 2W + 4 The answer is 2L + 2W = 42 and L = 2W + 4.