Create Systems

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Answer Key & Explanations

Create Systems · Grade 9 · Worksheet 2

1. Emma is analyzing two different investment plans for her savings. Plan A offers a fixed annual interest of $15 plus 5% of the amount invested. Plan B offers a fixed annual interest of $25 plus 3% of the amount invested. At what investment amount will both plans yield the same total annual interest, and what will that interest be? Write a system of equations to model this situation and solve. Answer: At $500 invested, both plans yield $40 in annual interest. Solution: Define variables. Let x = amount invested in dollars, and let y = total annual interest in dollars. Write equations for each plan.
   Full step-by-step solution

   Step 1: Define variables. Let x = amount invested in dollars, and let y = total annual interest in dollars. Step 2: Write equations for each plan. Plan A: y = 0.05x + 15 Plan B: y = 0.03x + 25 Step 3: Since both plans yield the same interest, set the equations equal to each other. 0.05x + 15 = 0.03x + 25 Step 4: Solve for x. 0.05x - 0.03x + 15 = 0.03x - 0.03x + 25 0.02x + 15 = 25 0.02x + 15 - 15 = 25 - 15 0.02x = 10 x = 10 / 0.02 x = 500 Step 5: Substitute x = 500 into either equation to find y. Using Plan A: y = 0.05(500) + 15 = 25 + 15 = 40 Using Plan B: y = 0.03(500) + 25 = 15 + 25 = 40 Step 6: Interpret the solution. When Emma invests $500, both plans give $40 in annual interest. The answer is: At $500 invested, both plans yield $40 in annual interest.

2. A theater sells tickets for $12 for adults and $8 for children. For a particular show, the theater sold 120 tickets and collected $1240. How many adult tickets were sold? Answer: 70 Solution: A = number of adult tickets C = number of child tickets 1. The total number of tickets sold is 120. So: A + C = 120 2.
   Full step-by-step solution

   Let's define variables for the number of adult tickets and child tickets. Let: A = number of adult tickets C = number of child tickets From the problem: 1. The total number of tickets sold is 120. So: A + C = 120 2. The total money collected is $1240. Adult tickets cost $12 each, child tickets cost $8 each. So: 12A + 8C = 1240 We now have the system of equations: Equation (1): A + C = 120 Equation (2): 12A + 8C = 1240 Step 1: Solve for one variable from Equation (1). From (1): C = 120 - A Step 2: Substitute C into Equation (2). 12A + 8(120 - A) = 1240 Step 3: Simplify and solve for A. 12A + 960 - 8A = 1240 (12A - 8A) + 960 = 1240 4A + 960 = 1240 Step 4: Subtract 960 from both sides. 4A = 1240 - 960 4A = 280 Step 5: Divide both sides by 4. A = 280 / 4 A = 70 So, the number of adult tickets sold is 70. Step 6: Check the answer. If A = 70, then C = 120 - 70 = 50. Money collected = 12*70 + 8*50 = 840 + 400 = 1240. This matches the problem statement. Final answer: 70 adult tickets.

3. Matiu is selling two types of handmade crafts at a market. He sold 8 of type A and 6 of type B for a total of $142. Type A sells for $4 more than type B. Write a system of equations to represent this situation. Answer: 8a + 6b = 142, a = b + 4 Solution: When creating systems of equations from word problems, we define variables for unknown quantities and translate the given relationships into mathematical statements.
   Full step-by-step solution

   When creating systems of equations from word problems, we define variables for unknown quantities and translate the given relationships into mathematical statements. The total revenue equation comes from multiplying quantity sold by price for each item and summing them. The price relationship equation expresses how one price compares to the other.

4. Olivia is comparing two different job offers. Company A offers a starting salary of $30,000 with an annual raise of $1,500. Company B offers a starting salary of $25,000 with an annual raise of $2,500. Write a system of equations to model the salary at each company after t years, and determine after how many years the salaries will be equal and what that salary will be. Answer: After 5 years, the salary will be $37,500. Solution: Define variables. Let t = number of years. Let S = salary after t years.
   Full step-by-step solution

   Step 1: Define variables. Let t = number of years. Let S = salary after t years. Step 2: Write equations for each company. Company A: S = 1500t + 30000 Company B: S = 2500t + 25000 Step 3: Set the equations equal to find when salaries are equal. 1500t + 30000 = 2500t + 25000 Step 4: Solve for t. 1500t + 30000 - 1500t = 2500t + 25000 - 1500t 30000 = 1000t + 25000 30000 - 25000 = 1000t + 25000 - 25000 5000 = 1000t t = 5000 / 1000 t = 5 Step 5: Find the salary at t = 5. Using Company A: S = 1500(5) + 30000 = 7500 + 30000 = 37500 Using Company B: S = 2500(5) + 25000 = 12500 + 25000 = 37500 Step 6: Interpret the solution. After 5 years, both companies will offer a salary of $37,500. The answer is after 5 years, the salary will be $37,500.

5. Hana and Matiu are saving money for a school trip. Hana has already saved $120 and plans to save $15 per week from her part-time job. Matiu has already saved $30 and plans to save $25 per week from his tutoring work. Write a system of equations to represent their total savings after w weeks, and determine after how many weeks they will have saved the same amount of money, and what that amount will be. Answer: After 9 weeks, both will have saved $255. Solution: Define variables. Let w = number of weeks. Let y = total savings in dollars.
   Full step-by-step solution

   Step 1: Define variables. Let w = number of weeks. Let y = total savings in dollars. Step 2: Write equations. Hana's savings: y = 15w + 120 Matiu's savings: y = 25w + 30 Step 3: Set the equations equal to find when savings are the same. 15w + 120 = 25w + 30 Step 4: Solve for w. 15w + 120 - 15w = 25w + 30 - 15w 120 = 10w + 30 120 - 30 = 10w + 30 - 30 90 = 10w w = 90 / 10 w = 9 Step 5: Find the total savings at w = 9. Using Hana's equation: y = 15(9) + 120 = 135 + 120 = 255 Using Matiu's equation: y = 25(9) + 30 = 225 + 30 = 255 Step 6: Interpret the solution. After 9 weeks, both Hana and Matiu will have saved $255 each. The answer is 9 weeks and $255.

6. Solve the system: 2x + 3y = 12 and x - y = 1 Answer: x = 3, y = 2 Solution: Use substitution method. From the second equation x - y = 1, we get x = y + 1 Substitute x = y + 1 into the first equation: 2(y + 1) + 3y = 12 Simplify: 2y + 2 + 3y = 12 Combine like terms: 5y + 2 = 12 Subtract 2 from both sides: 5y = 10 Divide by 5: y = 2 Substitute y = 2 into x = y + 1: x = 2…
   Full step-by-step solution

   Step 1: Use substitution method. From the second equation x - y = 1, we get x = y + 1 Step 2: Substitute x = y + 1 into the first equation: 2(y + 1) + 3y = 12 Step 3: Simplify: 2y + 2 + 3y = 12 Step 4: Combine like terms: 5y + 2 = 12 Step 5: Subtract 2 from both sides: 5y = 10 Step 6: Divide by 5: y = 2 Step 7: Substitute y = 2 into x = y + 1: x = 2 + 1 = 3 Step 8: Check: 2(3) + 3(2) = 6 + 6 = 12 ✓ and 3 - 2 = 1 ✓ The solution is x = 3, y = 2.

7. Matiu is selling two types of handmade crafts at a market. He sold 8 of type A and 6 of type B for a total of $124. Type A costs $4 more than type B. Write a system of equations to find the price of each type. Answer: a = 8, b = 4 Solution: When creating systems from word problems, identify what the variables represent, then translate each piece of information into an equation.
   Full step-by-step solution

   When creating systems from word problems, identify what the variables represent, then translate each piece of information into an equation. For example, if someone sold x of item P and y of item Q for total revenue R, that gives one equation. If one item costs C more than another, that gives a second equation relating the prices.