Systems of Equations

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Systems of Equations · Grade 8 · Worksheet 3

1. On a coordinate grid, a rectangular park is drawn with corners at (1, 1), (9, 1), (9, 7), and (1, 7). Two straight walking paths are planned inside the park. The first path goes from the midpoint of the left side (from (1,1) to (1,7)) to the midpoint of the right side (from (9,1) to (9,7)). The second path goes from the midpoint of the bottom side (from (1,1) to (9,1)) to the midpoint of the top side (from (1,7) to (9,7)). Find the coordinates of the point where these two paths intersect. Answer: (5, 4) Solution: Find the midpoint of the left side from (1,1) to (1,7). Midpoint = ((1+1)/2, (1+7)/2) = (2/2, 8/2) = (1, 4) Find the midpoint of the right side from (9,1) to (9,7).
   Full step-by-step solution

   Step 1: Find the midpoint of the left side from (1,1) to (1,7). Midpoint = ((1+1)/2, (1+7)/2) = (2/2, 8/2) = (1, 4) Step 2: Find the midpoint of the right side from (9,1) to (9,7). Midpoint = ((9+9)/2, (1+7)/2) = (18/2, 8/2) = (9, 4) Step 3: The first path goes from (1, 4) to (9, 4). This is a horizontal line. Its equation is y = 4. Step 4: Find the midpoint of the bottom side from (1,1) to (9,1). Midpoint = ((1+9)/2, (1+1)/2) = (10/2, 2/2) = (5, 1) Step 5: Find the midpoint of the top side from (1,7) to (9,7). Midpoint = ((1+9)/2, (7+7)/2) = (10/2, 14/2) = (5, 7) Step 6: The second path goes from (5, 1) to (5, 7). This is a vertical line. Its equation is x = 5. Step 7: The two paths intersect where x = 5 and y = 4. The intersection point is (5, 4). The answer is (5, 4).

2. 2x + 3y = 19; 5x - 2y = 14 Answer: x = 4, y = 3 Solution: Multiply the first equation by 2: 2(2x + 3y) = 2(19) → 4x + 6y = 38 Multiply the second equation by 3: 3(5x - 2y) = 3(14) → 15x - 6y = 42 Add the two equations: (4x + 6y) + (15x - 6y) = 38 + 42 → 19x = 80 → x = 80/19 = 4 Substitute x = 4 into the first equation: 2(4) + 3y = 19 → 8 + 3y = 19 → 3y…
   Full step-by-step solution

   Step 1: Multiply the first equation by 2: 2(2x + 3y) = 2(19) → 4x + 6y = 38 Step 2: Multiply the second equation by 3: 3(5x - 2y) = 3(14) → 15x - 6y = 42 Step 3: Add the two equations: (4x + 6y) + (15x - 6y) = 38 + 42 → 19x = 80 → x = 80/19 = 4 Step 4: Substitute x = 4 into the first equation: 2(4) + 3y = 19 → 8 + 3y = 19 → 3y = 11 → y = 11/3 = 3 The solution is x = 4, y = 3.

3. Olivia is selling tickets for the school play. Adult tickets cost $7 each and student tickets cost $5 each. She sold 35 tickets in total and collected $203. How many adult tickets and how many student tickets did she sell? Answer: 14 adult tickets and 21 student tickets Solution: Let a = number of adult tickets and s = number of student tickets.
   Full step-by-step solution

   Step 1: Let a = number of adult tickets and s = number of student tickets. Step 2: Equation for total tickets: a + s = 35 Step 3: Equation for total money: 7a + 5s = 203 Step 4: Solve the first equation for a: a = 35 - s Step 5: Substitute into the second equation: 7(35 - s) + 5s = 203 Step 6: Distribute: 245 - 7s + 5s = 203 Step 7: Combine like terms: 245 - 2s = 203 Step 8: Subtract 245 from both sides: -2s = -42 Step 9: Divide both sides by -2: s = 21 Step 10: Substitute back: a = 35 - 21 = 14 Step 11: Check: 7(14) + 5(21) = 98 + 105 = 203 The answer is 14 adult tickets and 21 student tickets.

4. Matiu is helping to organize the school library. He needs to sort books onto two types of shelves. Each small shelf holds 8 books and each large shelf holds 14 books. Matiu has a total of 26 shelves to fill and needs to shelve exactly 280 books. How many small shelves and how many large shelves should he use? Answer: 14 small shelves and 12 large shelves Solution: Let x = number of small shelves, and y = number of large shelves.
   Full step-by-step solution

   Step 1: Let x = number of small shelves, and y = number of large shelves. Step 2: Write the equation for the total number of shelves: x + y = 26 Step 3: Write the equation for the total number of books: 8x + 14y = 280 Step 4: Solve the first equation for x: x = 26 - y Step 5: Substitute into the second equation: 8(26 - y) + 14y = 280 Step 6: Simplify: 208 - 8y + 14y = 280 Step 7: Combine like terms: 208 + 6y = 280 Step 8: Subtract 208 from both sides: 6y = 72 Step 9: Divide both sides by 6: y = 12 Step 10: Substitute y = 12 back into x = 26 - y: x = 26 - 12 = 14 Step 11: Matiu needs 14 small shelves and 12 large shelves. Check: 14 small shelves hold 14 x 8 = 112 books. 12 large shelves hold 12 x 14 = 168 books. Total books = 112 + 168 = 280. Total shelves = 14 + 12 = 26. Correct! The answer is 14 small shelves and 12 large shelves.

5. 3x + 2y = 16; 2x - y = 1 Answer: x = 2, y = 3 Solution: Start with the system: 3x + 2y = 16 and 2x - y = 1 Multiply the second equation by 2 to match the y-coefficient: 2(2x - y) = 2(1) → 4x - 2y = 2 Add this to the first equation: (3x + 2y) + (4x - 2y) = 16 + 2 → 7x = 18 Solve for x: x = 18/7 = 2 Substitute x = 2 into the second equation: 2(2) - y =…
   Full step-by-step solution

   Step 1: Start with the system: 3x + 2y = 16 and 2x - y = 1 Step 2: Multiply the second equation by 2 to match the y-coefficient: 2(2x - y) = 2(1) → 4x - 2y = 2 Step 3: Add this to the first equation: (3x + 2y) + (4x - 2y) = 16 + 2 → 7x = 18 Step 4: Solve for x: x = 18/7 = 2 Step 5: Substitute x = 2 into the second equation: 2(2) - y = 1 → 4 - y = 1 Step 6: Solve for y: -y = 1 - 4 → -y = -3 → y = 3 Step 7: Check in first equation: 3(2) + 2(3) = 6 + 6 = 12 The solution is x = 2, y = 3.

6. Noah is helping to organize a community garden. He needs to buy two types of fertilizer: a standard mix that costs $9 per bag and an organic mix that costs $14 per bag. He has a total budget of $380 for fertilizer and needs to buy exactly 30 bags in total. How many bags of each type should Noah buy to spend exactly his budget? Answer: 8 bags of organic mix and 22 bags of standard mix Solution: Let x = number of standard mix bags and y = number of organic mix bags.
   Full step-by-step solution

   Let x = number of standard mix bags and y = number of organic mix bags. Equation 1 (total bags): x + y = 30 Equation 2 (total cost): 9x + 14y = 380 Solve Equation 1 for x: x = 30 - y Substitute into Equation 2: 9(30 - y) + 14y = 380 Simplify: 270 - 9y + 14y = 380 Combine like terms: 270 + 5y = 380 Subtract 270 from both sides: 5y = 110 Divide both sides by 5: y = 22 Substitute y = 22 into x = 30 - y: x = 30 - 22 = 8 Noah should buy 8 bags of standard mix and 22 bags of organic mix.

7. Liam is planning a school fundraiser and needs to order pizza for the event. He knows that cheese pizzas cost $12 each and pepperoni pizzas cost $15 each. If Liam orders a total of 8 pizzas and spends exactly $108, how many cheese pizzas and how many pepperoni pizzas did he order? Answer: 4 cheese pizzas and 4 pepperoni pizzas Solution: \( c \) = number of cheese pizzas \( p \) = number of pepperoni pizzas Total pizzas: \( c + p = 8 \) Total cost: \( 12c + 15p = 108 \) From \( c + p = 8 \), we get \( c = 8 - p \).
   Full step-by-step solution

   Let's solve this step by step. --- **Step 1: Define variables** Let \( c \) = number of cheese pizzas \( p \) = number of pepperoni pizzas --- **Step 2: Set up equations** From the problem: Total pizzas: \( c + p = 8 \) Total cost: \( 12c + 15p = 108 \) --- **Step 3: Solve for one variable** From \( c + p = 8 \), we get \( c = 8 - p \). Substitute into the cost equation: \( 12(8 - p) + 15p = 108 \) --- **Step 4: Simplify and solve for \( p \)** \( 96 - 12p + 15p = 108 \) \( 96 + 3p = 108 \) \( 3p = 108 - 96 \) \( 3p = 12 \) \( p = 4 \) --- **Step 5: Find \( c \)** \( c = 8 - p = 8 - 4 = 4 \) --- **Step 6: Check the solution** Cheese pizzas: \( 4 \times 12 = 48 \) dollars Pepperoni pizzas: \( 4 \times 15 = 60 \) dollars Total: \( 48 + 60 = 108 \) dollars Total pizzas: \( 4 + 4 = 8 \) pizzas Both conditions are satisfied. --- **Final answer:** Liam ordered 4 cheese pizzas and 4 pepperoni pizzas.