Infinite Solutions

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Infinite Solutions · Grade 8 · Worksheet 1

1. A local bakery sells cupcakes and cookies. For a school event, Liam bought 3 cupcakes and 2 cookies for a total of $9.50. Emma bought 1 cupcake and 4 cookies from the same bakery for $8.50. If the price of a cupcake is represented by 'c' and the price of a cookie is represented by 'k', which system of equations represents this situation? Answer: 3c + 2k = 9.50 and c + 4k = 8.50 Solution: - \( c \) = price of one cupcake - \( k \) = price of one cookie 3 cupcakes + 2 cookies → total $9.50 1 cupcake + 4 cookies → total $8.50 We need to write a system of equations.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the problem** We have two variables: - \( c \) = price of one cupcake - \( k \) = price of one cookie Liam's purchase: 3 cupcakes + 2 cookies → total $9.50 Emma's purchase: 1 cupcake + 4 cookies → total $8.50 We need to write a system of equations. --- **Step 2: Write Liam's equation** Liam bought 3 cupcakes → cost = \( 3 \times c \) Liam bought 2 cookies → cost = \( 2 \times k \) Total cost = \( 3c + 2k \) This equals $9.50. So: \[ 3c + 2k = 9.50 \] --- **Step 3: Write Emma's equation** Emma bought 1 cupcake → cost = \( 1 \times c \) Emma bought 4 cookies → cost = \( 4 \times k \) Total cost = \( c + 4k \) This equals $8.50. So: \[ c + 4k = 8.50 \] --- **Step 4: Write the system** The system of equations is: \[ \begin{cases} 3c + 2k = 9.50 \\ c + 4k = 8.50 \end{cases} \] --- **Final Answer:** 3c + 2k = 9.50 and c + 4k = 8.50

2. A school is organizing a field trip to the science museum. The total cost for the trip is $800, which includes a $200 booking fee plus $15 per student. The school wants to know how many students need to attend for the total cost per student to be exactly $20. Write and solve an equation to find the number of students. Answer: 40 Solution: Let x be the number of students. The total cost is $200 (booking fee) plus $15 per student: Total cost = 200 + 15x The cost per student is total cost divided by number of students: (200 + 15x)/x Set this equal to the desired cost per student of $20: (200 + 15x)/x = 20 Multiply both sides by x:…
   Full step-by-step solution

   Step 1: Let x be the number of students. Step 2: The total cost is $200 (booking fee) plus $15 per student: Total cost = 200 + 15x Step 3: The cost per student is total cost divided by number of students: (200 + 15x)/x Step 4: Set this equal to the desired cost per student of $20: (200 + 15x)/x = 20 Step 5: Multiply both sides by x: 200 + 15x = 20x Step 6: Subtract 15x from both sides: 200 = 5x Step 7: Divide both sides by 5: x = 40 Step 8: Check: Total cost = 200 + 15(40) = 200 + 600 = $800, Cost per student = 800/40 = $20 The answer is 40 students.

3. A local bakery sells cupcakes for $2.50 each and cookies for $1.25 each. Noah bought some cupcakes and cookies for a school event and spent $23.75 total. The number of cookies he bought was 3 more than twice the number of cupcakes. How many cupcakes did Noah buy? Answer: 4 Solution: c = number of cupcakes k = number of cookies Write equations from the problem. Cupcakes cost $2.50 each, cookies cost $1.25 each, total spent $23.75.
   Full step-by-step solution

   Let's define variables first. Let: c = number of cupcakes k = number of cookies Step 1: Write equations from the problem. From prices: Cupcakes cost $2.50 each, cookies cost $1.25 each, total spent $23.75. So: 2.50c + 1.25k = 23.75 From relationship between cookies and cupcakes: "Number of cookies was 3 more than twice the number of cupcakes" So: k = 2c + 3 Step 2: Substitute k from the second equation into the first equation. 2.50c + 1.25(2c + 3) = 23.75 Step 3: Simplify and solve for c. First, distribute 1.25: 2.50c + 1.25*2c + 1.25*3 = 23.75 2.50c + 2.50c + 3.75 = 23.75 Combine like terms: 5.00c + 3.75 = 23.75 Subtract 3.75 from both sides: 5.00c = 23.75 - 3.75 5.00c = 20.00 Divide both sides by 5: c = 20.00 / 5.00 c = 4 Step 4: Verify. If c = 4, then k = 2*4 + 3 = 11 cookies. Cost: 4 cupcakes * $2.50 = $10.00 11 cookies * $1.25 = $13.75 Total = $10.00 + $13.75 = $23.75 ✔ So Noah bought 4 cupcakes. ANSWER: 4

4. Hana is organizing a school fundraiser and is setting up an equation to calculate the total cost of buying supplies. She writes the equation 7(x + 12) = 7x + 84 to represent the cost in dollars for x packs of markers. Does this equation have one solution, no solution, or infinitely many solutions? Explain your reasoning. Answer: infinitely many solutions Solution: Start with the equation: 7(x + 12) = 7x + 84 Distribute the 7 on the left side: 7x + 84 = 7x + 84 Subtract 7x from both sides: 84 = 84 The statement 84 = 84 is always true, regardless of the value of x.
   Full step-by-step solution

   Step 1: Start with the equation: 7(x + 12) = 7x + 84 Step 2: Distribute the 7 on the left side: 7x + 84 = 7x + 84 Step 3: Subtract 7x from both sides: 84 = 84 Step 4: The statement 84 = 84 is always true, regardless of the value of x. Step 5: This means the equation is an identity, and it has infinitely many solutions. The answer is infinitely many solutions.

5. Mason is helping his aunt set up chairs for a community talent show. They need to arrange chairs in rows. The equation 7x + 12 = 7(x + 2) – 2 represents the number of chairs in each row, where x is the number of rows. How many solutions does this equation have? Answer: infinitely many solutions Solution: Start with the equation: 7x + 12 = 7(x + 2) – 2 Distribute the 7 on the right side: 7x + 12 = 7x + 14 – 2 Combine like terms on the right: 7x + 12 = 7x + 12 Subtract 7x from both sides: 12 = 12 The resulting statement is always true, regardless of the value of x.
   Full step-by-step solution

   Step 1: Start with the equation: 7x + 12 = 7(x + 2) – 2 Step 2: Distribute the 7 on the right side: 7x + 12 = 7x + 14 – 2 Step 3: Combine like terms on the right: 7x + 12 = 7x + 12 Step 4: Subtract 7x from both sides: 12 = 12 Step 5: The resulting statement is always true, regardless of the value of x. This means the equation has infinitely many solutions. Answer: infinitely many solutions

6. Mere is drawing a rectangle on a coordinate plane with vertices at (12, 10), (42, 10), (42, 25), and (12, 25). She draws a line from the midpoint of the left side to the midpoint of the right side. What is the equation of this line in slope-intercept form (y = mx + b)? Answer: y = 17.5 Solution: Identify the coordinates of the vertices. Left side: (12, 10) to (12, 25). Right side: (42, 10) to (42, 25).
   Full step-by-step solution

   Step 1: Identify the coordinates of the vertices. Left side: (12, 10) to (12, 25). Right side: (42, 10) to (42, 25). Step 2: Find the midpoint of the left side. Midpoint formula: ((x1+x2)/2, (y1+y2)/2). For left side: ((12+12)/2, (10+25)/2) = (12, 35/2) = (12, 17.5). Step 3: Find the midpoint of the right side. For right side: ((42+42)/2, (10+25)/2) = (42, 35/2) = (42, 17.5). Step 4: The line passes through (12, 17.5) and (42, 17.5). Both points have the same y-coordinate, so the slope is 0. Step 5: A horizontal line with slope 0 has equation y = constant. The constant is the y-coordinate, which is 17.5. Step 6: Therefore, the equation is y = 17.5. The answer is y = 17.5.

7. 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.