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Infinite Solutions

Grade 8 · Algebra · Worksheet 2

  1. Emma is planning a school fundraiser selling bracelets and keychains. She has $120 to spend on materials. Bracelet materials cost $2 per bracelet, and keychain materials cost $3 per keychain. If Emma wants to make the same number of each item and spend all her money, how many of each item should she make? Answer: ______________
  2. Emma is helping her cousin with a lemonade stand. They decide to charge $3 for each cup of lemonade. Emma's cousin claims that the amount of money they earn, in dollars, can be found using the equation 3x + 5 = 3(x + 2) - 1, where x is the number of cups sold. Emma thinks this equation will always give the correct total, no matter how many cups they sell. Is Emma's cousin correct? Does the equation have one solution, no solution, or infinitely many solutions? Explain your reasoning. Answer: ______________
  3. A triangle is drawn on a coordinate plane with vertices at A(2, 3), B(8, 3), and C(5, 7). A line is drawn from vertex C to the midpoint of side AB. What are the coordinates of this midpoint? Answer: ______________
  4. Emma is planning a school fundraiser and needs to order custom t-shirts. The printing company charges a flat setup fee of $75 plus $8 per shirt. Emma's budget for the t-shirts is $475. How many t-shirts can she order without exceeding her budget? Answer: ______________
  5. Mere is helping her aunt prepare goodie bags for a school fair. They have a large box of lollipops and a large box of stickers. Mere notices that when they put the same number of lollipops into each of 10 bags, they have 8 lollipops left over. When they put the same number of stickers into each of those 10 bags, they also have 8 stickers left over. Mere's aunt says, 'If we combine all the lollipops and stickers and then split them equally among the 10 bags, we will still have exactly 8 items left over.' Mere writes the equation 10(l + s) + 8 = 10l + 10s + 8, where l is the number of lollipops per bag and s is the number of stickers per bag. How many solutions does this equation have—one solution, no solution, or infinitely many solutions? Explain your reasoning. Answer: ______________
  6. Emma is mixing a special cleaning solution for her science experiment. She needs to combine Solution A and Solution B in a specific ratio. When she mixes 2 parts of Solution A with 3 parts of Solution B, she gets 500 milliliters of the perfect mixture. If she wants to maintain the same ratio but make 750 milliliters total, how many milliliters of Solution A should she use? Answer: ______________
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Answer Key & Explanations

Infinite Solutions · Grade 8 · Worksheet 2

  1. Emma is planning a school fundraiser selling bracelets and keychains. She has $120 to spend on materials. Bracelet materials cost $2 per bracelet, and keychain materials cost $3 per keychain. If Emma wants to make the same number of each item and spend all her money, how many of each item should she make? Answer: 24 Solution: Let x represent the number of bracelets and the number of keychains (since she makes the same number of each). The cost for bracelets is 2x dollars. The cost for keychains is 3x dollars.
    Full step-by-step solution

    Step 1: Let x represent the number of bracelets and the number of keychains (since she makes the same number of each). Step 2: The cost for bracelets is 2x dollars. Step 3: The cost for keychains is 3x dollars. Step 4: The total cost equation is: 2x + 3x = 120 Step 5: Combine like terms: 5x = 120 Step 6: Divide both sides by 5: x = 120 ÷ 5 Step 7: x = 24 Emma should make 24 bracelets and 24 keychains.

  2. Emma is helping her cousin with a lemonade stand. They decide to charge $3 for each cup of lemonade. Emma's cousin claims that the amount of money they earn, in dollars, can be found using the equation 3x + 5 = 3(x + 2) - 1, where x is the number of cups sold. Emma thinks this equation will always give the correct total, no matter how many cups they sell. Is Emma's cousin correct? Does the equation have one solution, no solution, or infinitely many solutions? Explain your reasoning. Answer: infinitely many solutions Solution: Write the equation: 3x + 5 = 3(x + 2) - 1. Distribute on the right side: 3(x + 2) - 1 = 3x + 6 - 1 = 3x + 5. Now the equation is 3x + 5 = 3x + 5.
    Full step-by-step solution

    Step 1: Write the equation: 3x + 5 = 3(x + 2) - 1. Step 2: Distribute on the right side: 3(x + 2) - 1 = 3x + 6 - 1 = 3x + 5. Step 3: Now the equation is 3x + 5 = 3x + 5. Step 4: Subtract 3x from both sides: 5 = 5. Step 5: This is a true statement that does not depend on x. Therefore, the equation is an identity. The equation has infinitely many solutions, meaning it is true for any value of x (any number of cups sold). Emma's cousin is correct that the equation always gives the correct total.

  3. A triangle is drawn on a coordinate plane with vertices at A(2, 3), B(8, 3), and C(5, 7). A line is drawn from vertex C to the midpoint of side AB. What are the coordinates of this midpoint? Answer: (5, 3) Solution: Identify the endpoints of side AB: A(2, 3) and B(8, 3) Find the x-coordinate of the midpoint: (2 + 8) ÷ 2 = 10 ÷ 2 = 5 Find the y-coordinate of the midpoint: (3 + 3) ÷ 2 = 6 ÷ 2 = 3 Combine the coordinates: (5, 3) The midpoint of side AB is (5, 3).
    Full step-by-step solution

    Step 1: Identify the endpoints of side AB: A(2, 3) and B(8, 3) Step 2: Find the x-coordinate of the midpoint: (2 + 8) ÷ 2 = 10 ÷ 2 = 5 Step 3: Find the y-coordinate of the midpoint: (3 + 3) ÷ 2 = 6 ÷ 2 = 3 Step 4: Combine the coordinates: (5, 3) The midpoint of side AB is (5, 3).

  4. Emma is planning a school fundraiser and needs to order custom t-shirts. The printing company charges a flat setup fee of $75 plus $8 per shirt. Emma's budget for the t-shirts is $475. How many t-shirts can she order without exceeding her budget? Answer: 50 Solution: Let x represent the number of t-shirts Emma can order. The total cost is the setup fee plus the cost per shirt: 75 + 8x Set up the equation where total cost equals budget: 75 + 8x = 475 Subtract 75 from both sides: 8x = 400 Divide both sides by 8: x = 50 Emma can order 50 t-shirts without…
    Full step-by-step solution

    Step 1: Let x represent the number of t-shirts Emma can order. Step 2: The total cost is the setup fee plus the cost per shirt: 75 + 8x Step 3: Set up the equation where total cost equals budget: 75 + 8x = 475 Step 4: Subtract 75 from both sides: 8x = 400 Step 5: Divide both sides by 8: x = 50 Step 6: Emma can order 50 t-shirts without exceeding her budget.

  5. Mere is helping her aunt prepare goodie bags for a school fair. They have a large box of lollipops and a large box of stickers. Mere notices that when they put the same number of lollipops into each of 10 bags, they have 8 lollipops left over. When they put the same number of stickers into each of those 10 bags, they also have 8 stickers left over. Mere's aunt says, 'If we combine all the lollipops and stickers and then split them equally among the 10 bags, we will still have exactly 8 items left over.' Mere writes the equation 10(l + s) + 8 = 10l + 10s + 8, where l is the number of lollipops per bag and s is the number of stickers per bag. How many solutions does this equation have—one solution, no solution, or infinitely many solutions? Explain your reasoning. Answer: infinitely many solutions Solution: Start with the equation 10(l + s) + 8 = 10l + 10s + 8. Expand the left side: 10(l + s) + 8 = 10l + 10s + 8. The equation becomes 10l + 10s + 8 = 10l + 10s + 8.
    Full step-by-step solution

    Step 1: Start with the equation 10(l + s) + 8 = 10l + 10s + 8. Step 2: Expand the left side: 10(l + s) + 8 = 10l + 10s + 8. Step 3: The equation becomes 10l + 10s + 8 = 10l + 10s + 8. Step 4: Subtract 10l from both sides: 10s + 8 = 10s + 8. Step 5: Subtract 10s from both sides: 8 = 8. Step 6: This is a true statement with no variables left, meaning the equation is an identity. Step 7: An identity has infinitely many solutions because any values of l and s will satisfy the original equation. The answer is infinitely many solutions.

  6. Emma is mixing a special cleaning solution for her science experiment. She needs to combine Solution A and Solution B in a specific ratio. When she mixes 2 parts of Solution A with 3 parts of Solution B, she gets 500 milliliters of the perfect mixture. If she wants to maintain the same ratio but make 750 milliliters total, how many milliliters of Solution A should she use? Answer: 300 Solution: Step 1: The ratio is 2 parts Solution A to 3 parts Solution B, so total parts = 2 + 3 = 5 parts Step 2: In the original mixture, Solution A represents 2/5 of the total Step 3: For the new mixture of 750 ml, Solution A needed = (2/5) × 750 Step 4: Calculate: (2/5) × 750 = 2 × 150 = 300 Step 5:…
    Full step-by-step solution

    Step 1: The ratio is 2 parts Solution A to 3 parts Solution B, so total parts = 2 + 3 = 5 parts Step 2: In the original mixture, Solution A represents 2/5 of the total Step 3: For the new mixture of 750 ml, Solution A needed = (2/5) × 750 Step 4: Calculate: (2/5) × 750 = 2 × 150 = 300 Step 5: Emma should use 300 milliliters of Solution A Step 6: Verify: If Solution A = 300 ml, then Solution B = 750 - 300 = 450 ml Step 7: Check ratio: 300:450 = 2:3 (divide both by 150) The answer is 300 milliliters.