Systems of Equations

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Systems of Equations · Grade 8 · Worksheet 2

1. Emma is in charge of buying snacks for the school movie night. She needs to buy bags of popcorn and boxes of candy. Each bag of popcorn costs $3, and each box of candy costs $5. Emma buys a total of 21 items and spends exactly $79. How many bags of popcorn and how many boxes of candy did she buy? Answer: 13 bags of popcorn and 8 boxes of candy Solution: Let p = number of bags of popcorn, and c = number of boxes of candy. Equation 1 (total items): p + c = 21 Equation 2 (total cost): 3p + 5c = 79 Solve by substitution.
   Full step-by-step solution

   Let p = number of bags of popcorn, and c = number of boxes of candy. Equation 1 (total items): p + c = 21 Equation 2 (total cost): 3p + 5c = 79 Solve by substitution. From Equation 1: p = 21 - c Substitute into Equation 2: 3(21 - c) + 5c = 79 Simplify: 63 - 3c + 5c = 79 Combine like terms: 63 + 2c = 79 Subtract 63 from both sides: 2c = 16 Divide by 2: c = 8 Now find p: p = 21 - 8 = 13 Emma bought 13 bags of popcorn and 8 boxes of candy.

2. Matiu is designing a kite on a coordinate grid. The kite is a quadrilateral with vertices at (2, 18), (14, 26), (26, 18), and (14, 10). He wants to add a decorative string that runs in a straight line from the midpoint of the top edge (connecting (2,18) to (14,26)) to the midpoint of the bottom edge (connecting (14,10) to (26,18)). Another string runs along the vertical line x = 14. At what coordinates do these two strings intersect? Answer: (14, 18) Solution: Find the midpoint of the top edge from (2,18) to (14,26). Midpoint = ((2+14)/2, (18+26)/2) = (16/2, 44/2) = (8, 22). Step 2: Find the midpoint of the bottom edge from (14,10) to (26,18).
   Full step-by-step solution

   Step 1: Find the midpoint of the top edge from (2,18) to (14,26). Midpoint = ((2+14)/2, (18+26)/2) = (16/2, 44/2) = (8, 22). Step 2: Find the midpoint of the bottom edge from (14,10) to (26,18). Midpoint = ((14+26)/2, (10+18)/2) = (40/2, 28/2) = (20, 14). Step 3: The decorative string connects (8,22) and (20,14). This is a straight line. The other string is the vertical line x = 14. Step 4: The intersection point has an x-coordinate of 14. The y-coordinate is on the line connecting (8,22) and (20,14). Step 5: The line connecting the midpoints is horizontal because the y-coordinates change from 22 to 14 over the x-coordinates from 8 to 20. At x = 14, the y-coordinate is exactly halfway between 22 and 14 because 14 is halfway between 8 and 20. Halfway between 22 and 14 is (22+14)/2 = 36/2 = 18. Step 6: Therefore, the intersection point is (14, 18).

3. Noah is in charge of ordering sandwiches and salads for a school picnic. Each sandwich costs $6 and each salad costs $11. He orders a total of 26 items and spends exactly $206. How many sandwiches and how many salads did Noah order? Answer: 16 sandwiches and 10 salads Solution: Let s = number of sandwiches, and a = number of salads. Equation 1 (total items): s + a = 26 Equation 2 (total cost): 6s + 11a = 206 Solve by substitution: From equation 1, s = 26 - a.
   Full step-by-step solution

   Let s = number of sandwiches, and a = number of salads. Equation 1 (total items): s + a = 26 Equation 2 (total cost): 6s + 11a = 206 Solve by substitution: From equation 1, s = 26 - a. Substitute into equation 2: 6(26 - a) + 11a = 206 Simplify: 156 - 6a + 11a = 206 Combine like terms: 156 + 5a = 206 Subtract 156 from both sides: 5a = 50 Divide by 5: a = 10 Now find s: s = 26 - 10 = 16 Noah ordered 16 sandwiches and 10 salads.

4. Emma is organizing a school art show and needs to frame two different sizes of artwork. The small frames cost $8 each and the large frames cost $15 each. She has a budget of $200 for frames and needs exactly 18 frames total. How many small frames and how many large frames should Emma order to spend exactly her budget? Answer: 10 small frames and 8 large frames Solution: Systems of linear equations help solve problems where you have multiple constraints involving the same variables.
   Full step-by-step solution

   Systems of linear equations help solve problems where you have multiple constraints involving the same variables. For example, if you were buying two types of books with different prices and knew both the total number of books and total cost, you could set up equations to find how many of each type.

5. Emma is organizing a school field trip to the science museum. She needs to arrange transportation using vans and buses. Each van can carry 8 students and costs $40 to rent, while each bus can carry 40 students and costs $120 to rent. There are 200 students going on the trip, and Emma has a total budget of $600 for transportation. How many vans and buses should she use to transport all students while staying within her budget? Answer: 5 vans and 4 buses Solution: One equation typically deals with quantities (like number of people or items), while another deals with costs or other measurements. The solution must satisfy both equations simultaneously.
   Full step-by-step solution

   When solving systems of linear equations with real-world constraints, it's helpful to identify what each equation represents. One equation typically deals with quantities (like number of people or items), while another deals with costs or other measurements. The solution must satisfy both equations simultaneously.

6. Emma is organizing a school art show and needs to frame her artwork. She has two types of frames: small frames that cost $6 each and large frames that cost $10 each. She bought a total of 15 frames and spent exactly $118. How many small frames and how many large frames did Emma buy? Answer: 8 Solution: Let s = number of small frames and l = number of large frames.
   Full step-by-step solution

   Let s = number of small frames and l = number of large frames. From the total frames: s + l = 15 From the total cost: 6s + 10l = 118 Solve the first equation for s: s = 15 - l Substitute into the second equation: 6(15 - l) + 10l = 118 90 - 6l + 10l = 118 90 + 4l = 118 4l = 118 - 90 4l = 28 l = 7 Now find s: s = 15 - 7 = 8 Emma bought 8 small frames and 7 large frames. The question asks for the number of small frames, so the answer is 8.

7. Maya is organizing a school art show and needs to frame her artwork. She has two types of frames: small frames that cost $4 each and large frames that cost $7 each. Maya bought a total of 15 frames and spent exactly $90. How many small frames and how many large frames did Maya buy? Answer: 5 small frames and 10 large frames Solution: Let x = number of small frames, y = number of large frames Write the equation for total frames: x + y = 15 Write the equation for total cost: 4x + 7y = 90 Solve the first equation for x: x = 15 - y Substitute into the second equation: 4(15 - y) + 7y = 90 Simplify: 60 - 4y + 7y = 90 Combine like…
   Full step-by-step solution

   Step 1: Let x = number of small frames, y = number of large frames Step 2: Write the equation for total frames: x + y = 15 Step 3: Write the equation for total cost: 4x + 7y = 90 Step 4: Solve the first equation for x: x = 15 - y Step 5: Substitute into the second equation: 4(15 - y) + 7y = 90 Step 6: Simplify: 60 - 4y + 7y = 90 Step 7: Combine like terms: 60 + 3y = 90 Step 8: Subtract 60 from both sides: 3y = 30 Step 9: Divide by 3: y = 10 Step 10: Substitute back to find x: x = 15 - 10 = 5 Step 11: Maya bought 5 small frames and 10 large frames.