Infinite Solutions

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Infinite Solutions · Grade 8 · Worksheet 2

1. A chemist is preparing a solution and needs to determine when two different mixing processes will create the same concentration. The concentration after x minutes is given by the equation 4(3x - 7) + 12 = kx - 16. For what value of k will this equation have infinitely many solutions, representing identical concentration at all times? Answer: 12 Solution: Expand the left side: 4(3x - 7) + 12 = 12x - 28 + 12 = 12x - 16 The equation becomes: 12x - 16 = kx - 16 For infinitely many solutions, the coefficients of x must be equal and the constants must be equal Compare x coefficients: 12 = k Compare constants: -16 = -16 (this is already true)…
   Full step-by-step solution

   Step 1: Expand the left side: 4(3x - 7) + 12 = 12x - 28 + 12 = 12x - 16 Step 2: The equation becomes: 12x - 16 = kx - 16 Step 3: For infinitely many solutions, the coefficients of x must be equal and the constants must be equal Step 4: Compare x coefficients: 12 = k Step 5: Compare constants: -16 = -16 (this is already true) Step 6: Therefore, k must equal 12 for the equation to have infinitely many solutions The answer is 12.

2. A school is planning a field trip and needs to rent buses. Each bus can hold 40 students. The total cost for renting buses is $500 plus $25 per bus. If the total cost for the trip is $900, how many buses did the school rent? Answer: 16 Solution: Let \( b \) = number of buses rented.
   Full step-by-step solution

   Let's break this down step by step. --- **Step 1: Define the variables** Let \( b \) = number of buses rented. --- **Step 2: Write the cost equation** The problem says: - Fixed cost: $500 - Plus $25 per bus So total cost = \( 500 + 25 \times b \) We know total cost is $900, so: \[ 500 + 25b = 900 \] --- **Step 3: Isolate the term with \( b \)** Subtract 500 from both sides: \[ 25b = 900 - 500 \] \[ 25b = 400 \] --- **Step 4: Solve for \( b \)** Divide both sides by 25: \[ b = 400 / 25 \] \[ b = 16 \] --- **Step 5: Interpret the result** The school rented 16 buses. --- **Final answer:** 16

3. 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.

4. Emma is planning a school fundraiser and needs to order custom t-shirts. The printing company charges a flat setup fee of $50 plus $8 per shirt. Emma has a budget of $250 for the t-shirts. Write an equation to represent this situation, then solve it to find how many t-shirts she can order without exceeding her budget. Answer: 25 Solution: Let x represent the number of t-shirts Emma can order. The total cost is the setup fee plus the cost per shirt: 50 + 8x Set this equal to her budget: 50 + 8x = 250 Subtract 50 from both sides: 8x = 200 Divide both sides by 8: x = 25 Emma can order 25 t-shirts without exceeding her budget.
   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: 50 + 8x Step 3: Set this equal to her budget: 50 + 8x = 250 Step 4: Subtract 50 from both sides: 8x = 200 Step 5: Divide both sides by 8: x = 25 Emma can order 25 t-shirts without exceeding her budget.

5. 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.

6. Liam is designing a community garden with rectangular plots. He needs to determine the dimensions where the length is 5 meters more than twice the width. The total area of each plot should be 88 square meters. What are the dimensions (width and length) of each garden plot? Answer: width = 5.5 meters, length = 16 meters Solution: Let the width = w meters. The length is 5 meters more than twice the width, so: length = 2w + 5.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Define variables** Let the width = w meters. The length is 5 meters more than twice the width, so: length = 2w + 5. --- **Step 2: Write the area equation** Area = length × width 88 = (2w + 5) × w --- **Step 3: Expand and rearrange** 88 = 2w^2 + 5w Bring all terms to one side: 2w^2 + 5w - 88 = 0 --- **Step 4: Solve the quadratic equation** We can use the quadratic formula: w = [ -b ± sqrt(b^2 - 4ac) ] / (2a) Here a = 2, b = 5, c = -88. Discriminant: b^2 - 4ac = 25 - 4(2)(-88) = 25 + 704 = 729. sqrt(729) = 27. So: w = [ -5 ± 27 ] / (4). --- **Step 5: Find the two possible solutions** First: w = ( -5 + 27 ) / 4 = 22 / 4 = 5.5 Second: w = ( -5 - 27 ) / 4 = -32 / 4 = -8 --- **Step 6: Interpret the solutions** Width cannot be negative, so w = 5.5 meters. --- **Step 7: Find the length** length = 2w + 5 = 2(5.5) + 5 = 11 + 5 = 16 meters. --- **Step 8: Check** Area = 16 × 5.5 = 88 square meters. Length (16) is indeed 5 more than twice the width (2 × 5.5 = 11, plus 5 is 16). --- **Final answer:** width = 5.5 meters, length = 16 meters

7. Noah is helping his aunt plan a rectangular garden in her backyard. She wants the length of the garden to be 6 meters more than one-third of the width. Noah wrote the equation 2w + 2((1/3)w + 6) = 2w + (2/3)w + 12 to represent the perimeter. After simplifying both sides, he noticed the equation becomes (8/3)w + 12 = (8/3)w + 12. How many possible width values satisfy this equation? Explain what this means about the dimensions of the garden. Answer: Infinitely many solutions Solution: The equation is (8/3)w + 12 = (8/3)w + 12. Subtract (8/3)w from both sides: (8/3)w + 12 - (8/3)w = (8/3)w + 12 - (8/3)w This simplifies to 12 = 12.
   Full step-by-step solution

   Step 1: The equation is (8/3)w + 12 = (8/3)w + 12. Step 2: Subtract (8/3)w from both sides: (8/3)w + 12 - (8/3)w = (8/3)w + 12 - (8/3)w Step 3: This simplifies to 12 = 12. Step 4: Since 12 = 12 is always true (a true statement with no variable), the equation is an identity. Step 5: An identity means the equation holds true for any value of w. Step 6: Therefore, the equation has infinitely many solutions. This means any positive width will satisfy the perimeter equation, as long as the length is 6 more than one-third of that width. The dimensions are not uniquely determined by this equation alone. The answer is: infinitely many solutions.