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Infinite Solutions

Grade 8 · Algebra · Worksheet 1

  1. Liam is designing a rectangular garden with a perimeter of 48 feet. The length of the garden is 4 feet more than twice its width. Write an equation to represent this situation and find the dimensions of the garden. Answer: ______________
  2. Isabella is volunteering at a local animal shelter. She is tasked with organizing the donation of dog food bags. The shelter received a donation of 12 bags of dry dog food and 18 cans of wet dog food. Isabella wants to create identical care packages for the dogs, each containing the same number of bags of dry food and the same number of cans of wet food, using all the donated food. She claims that if she makes 6 care packages, the equation 12x + 18 = 6(2x + 3) describes the situation, where x represents the number of bags of dry food per package. How many solutions does this equation have, and what does that tell you about Isabella's plan? Answer: ______________
  3. Mere is helping her aunt set up chairs for a school concert. They need to arrange chairs in rows with the same number of chairs in each row. Mere notices that if they set up 4 chairs in each row, they will have 6 chairs left over. If they set up 6 chairs in each row, they will have 2 chairs left over. Mere writes the equation 4x + 6 = 6x + 2, where x represents the number of rows. Does this equation have one solution, no solution, or infinitely many solutions? Explain your reasoning. Answer: ______________
  4. 3(2x + 4) - 2(3x + 6) = ? Answer: ______________
  5. A triangle is drawn on a coordinate plane with vertices at A(1, 2), B(5, 2), and C(3, 6). A line is drawn from point A to the midpoint of side BC. What are the coordinates of this midpoint? Answer: ______________
  6. Emma is organizing a school fundraiser and needs to buy supplies for making bracelets and keychains. She has a budget of $200 to spend on beads and key rings. Beads cost $4 per pack and key rings cost $2 each. Emma wants to buy the same number of bead packs as key rings. How many of each item can she buy to spend exactly her $200 budget? Answer: ______________
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Answer Key & Explanations

Infinite Solutions · Grade 8 · Worksheet 1

  1. Liam is designing a rectangular garden with a perimeter of 48 feet. The length of the garden is 4 feet more than twice its width. Write an equation to represent this situation and find the dimensions of the garden. Answer: width = 6.67 feet, length = 17.33 feet Solution: width = w (in feet) length = l (in feet) Perimeter = 48 feet 2 * length + 2 * width = 48 2l + 2w = 48 Length is 4 feet more than twice the width: l = 2w + 4 Substitute the length expression into the perimeter equation From l = 2w + 4, substitute into 2l + 2w = 48: 2(2w + 4) + 2w = 48 2(2w + 4) +…
    Full step-by-step solution

    Let's go step-by-step. --- **Step 1: Define variables** Let width = w (in feet) length = l (in feet) --- **Step 2: Translate the problem into equations** We are told: Perimeter = 48 feet Perimeter formula for a rectangle: 2 * length + 2 * width = 48 So: 2l + 2w = 48 Also: Length is 4 feet more than twice the width: l = 2w + 4 --- **Step 3: Substitute the length expression into the perimeter equation** From l = 2w + 4, substitute into 2l + 2w = 48: 2(2w + 4) + 2w = 48 --- **Step 4: Simplify and solve for w** 2(2w + 4) + 2w = 48 4w + 8 + 2w = 48 6w + 8 = 48 6w = 48 - 8 6w = 40 w = 40 / 6 w = 20 / 3 w ≈ 6.6667 feet --- **Step 5: Find length l** l = 2w + 4 l = 2*(20/3) + 4 l = 40/3 + 12/3 l = 52/3 l ≈ 17.3333 feet --- **Step 6: Final answer** Width = 20/3 feet ≈ 6.67 feet Length = 52/3 feet ≈ 17.33 feet

  2. Isabella is volunteering at a local animal shelter. She is tasked with organizing the donation of dog food bags. The shelter received a donation of 12 bags of dry dog food and 18 cans of wet dog food. Isabella wants to create identical care packages for the dogs, each containing the same number of bags of dry food and the same number of cans of wet food, using all the donated food. She claims that if she makes 6 care packages, the equation 12x + 18 = 6(2x + 3) describes the situation, where x represents the number of bags of dry food per package. How many solutions does this equation have, and what does that tell you about Isabella's plan? Answer: Infinitely many solutions Solution: Write the equation: 12x + 18 = 6(2x + 3). Distribute the 6 on the right side: 6(2x + 3) = 6 * 2x + 6 * 3 = 12x + 18. The equation becomes: 12x + 18 = 12x + 18.
    Full step-by-step solution

    Step 1: Write the equation: 12x + 18 = 6(2x + 3). Step 2: Distribute the 6 on the right side: 6(2x + 3) = 6 * 2x + 6 * 3 = 12x + 18. Step 3: The equation becomes: 12x + 18 = 12x + 18. Step 4: Subtract 12x from both sides: 12x - 12x + 18 = 12x - 12x + 18, which simplifies to 18 = 18. Step 5: Since 18 = 18 is always true, the equation has infinitely many solutions. Step 6: This means Isabella's plan works for any number of bags per package as long as the total matches. For 6 packages, each package could have 2 bags of dry food (since 12 bags / 6 packages = 2) and 3 cans of wet food (since 18 cans / 6 packages = 3), which fits the equation. The equation having infinitely many solutions confirms that her plan is consistent.

  3. Mere is helping her aunt set up chairs for a school concert. They need to arrange chairs in rows with the same number of chairs in each row. Mere notices that if they set up 4 chairs in each row, they will have 6 chairs left over. If they set up 6 chairs in each row, they will have 2 chairs left over. Mere writes the equation 4x + 6 = 6x + 2, where x represents the number of rows. Does this equation have one solution, no solution, or infinitely many solutions? Explain your reasoning. Answer: one solution Solution: Start with the equation: 4x + 6 = 6x + 2 Subtract 4x from both sides to get the variable terms together: 4x + 6 - 4x = 6x + 2 - 4x, which simplifies to 6 = 2x + 2 Subtract 2 from both sides: 6 - 2 = 2x + 2 - 2, which gives 4 = 2x Divide both sides by 2: 4 / 2 = 2x / 2, so x = 2 Since we found…
    Full step-by-step solution

    Step 1: Start with the equation: 4x + 6 = 6x + 2 Step 2: Subtract 4x from both sides to get the variable terms together: 4x + 6 - 4x = 6x + 2 - 4x, which simplifies to 6 = 2x + 2 Step 3: Subtract 2 from both sides: 6 - 2 = 2x + 2 - 2, which gives 4 = 2x Step 4: Divide both sides by 2: 4 / 2 = 2x / 2, so x = 2 Step 5: Since we found exactly one value for x that makes the equation true, the equation has one solution. The answer is one solution.

  4. 3(2x + 4) - 2(3x + 6) = ? Answer: 0 Solution: Distribute the 3: 3(2x + 4) = 6x + 12 Distribute the -2: -2(3x + 6) = -6x - 12 Combine the expressions: 6x + 12 - 6x - 12 Combine like terms: (6x - 6x) + (12 - 12) = 0x + 0 Simplify: 0x + 0 = 0 The answer is 0.
    Full step-by-step solution

    Step 1: Distribute the 3: 3(2x + 4) = 6x + 12 Step 2: Distribute the -2: -2(3x + 6) = -6x - 12 Step 3: Combine the expressions: 6x + 12 - 6x - 12 Step 4: Combine like terms: (6x - 6x) + (12 - 12) = 0x + 0 Step 5: Simplify: 0x + 0 = 0 The answer is 0.

  5. A triangle is drawn on a coordinate plane with vertices at A(1, 2), B(5, 2), and C(3, 6). A line is drawn from point A to the midpoint of side BC. What are the coordinates of this midpoint? Answer: (4, 4) Solution: Identify the endpoints of side BC: B(5, 2) and C(3, 6). Calculate the x-coordinate of the midpoint: (5 + 3) ÷ 2 = 8 ÷ 2 = 4. Calculate the y-coordinate of the midpoint: (2 + 6) ÷ 2 = 8 ÷ 2 = 4.
    Full step-by-step solution

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

  6. Emma is organizing a school fundraiser and needs to buy supplies for making bracelets and keychains. She has a budget of $200 to spend on beads and key rings. Beads cost $4 per pack and key rings cost $2 each. Emma wants to buy the same number of bead packs as key rings. How many of each item can she buy to spend exactly her $200 budget? Answer: 33 Solution: Let x represent the number of bead packs and key rings (since she wants equal quantities) Cost of bead packs: 4x dollars Cost of key rings: 2x dollars Total cost equation: 4x + 2x = 200 Combine like terms: 6x = 200 Divide both sides by 6: x = 200 ÷ 6 Calculate: x = 33.33 Since we can't buy…
    Full step-by-step solution

    Step 1: Let x represent the number of bead packs and key rings (since she wants equal quantities) Step 2: Cost of bead packs: 4x dollars Step 3: Cost of key rings: 2x dollars Step 4: Total cost equation: 4x + 2x = 200 Step 5: Combine like terms: 6x = 200 Step 6: Divide both sides by 6: x = 200 ÷ 6 Step 7: Calculate: x = 33.33 Step 8: Since we can't buy partial items, we check: 33 bead packs cost 33 × 4 = $132, 33 key rings cost 33 × 2 = $66, total = $132 + $66 = $198 Step 9: With $2 left, we cannot buy another complete set (which would cost $6) Step 10: Therefore, Emma can buy 33 of each item The answer is 33.