Systems Word Problems

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Answer Key & Explanations

Systems Word Problems · Grade 8 · Worksheet 1

1. Matiu and Mere together have $124. Matiu has $16 more than twice the amount Mere has. How much money does each person have? Answer: Matiu has $88, Mere has $36 Solution: Define variables. Let x = amount Matiu has, y = amount Mere has. Write equations.
   Full step-by-step solution

   Step 1: Define variables. Let x = amount Matiu has, y = amount Mere has. Step 2: Write equations. Total money: x + y = 124. Relationship: x = 2y + 16. Step 3: Substitute the second equation into the first: (2y + 16) + y = 124. Step 4: Simplify: 3y + 16 = 124. Step 5: Subtract 16 from both sides: 3y = 108. Step 6: Divide by 3: y = 36. Step 7: Substitute y = 36 into x = 2y + 16: x = 2(36) + 16 = 72 + 16 = 88. Step 8: Check: 88 + 36 = 124, and 88 = 2(36) + 16 = 72 + 16 = 88. Correct. The answer is Matiu has $88 and Mere has $36.

2. Liam is planning a school fundraiser and needs to figure out how many boxes of cookies and candy bars to sell. He knows that cookies sell for $3 per box and candy bars sell for $2 each. His goal is to raise exactly $120. He also has a limit on storage space and can only hold a total of 50 items. How many boxes of cookies and how many candy bars should Liam plan to sell? Answer: 20 boxes of cookies and 30 candy bars Solution: Let c = number of boxes of cookies Let b = number of candy bars 1. Each cookie box costs $3, each candy bar costs $2, total money = $120 Equation 1: 3c + 2b = 120 2.
   Full step-by-step solution

   Let's define variables: Let c = number of boxes of cookies Let b = number of candy bars From the problem: 1. Each cookie box costs $3, each candy bar costs $2, total money = $120 Equation 1: 3c + 2b = 120 2. Total items cannot exceed 50 Equation 2: c + b = 50 We now solve this system of equations. Step 1: Solve Equation 2 for one variable c + b = 50 c = 50 - b Step 2: Substitute into Equation 1 3(50 - b) + 2b = 120 Step 3: Simplify and solve for b 150 - 3b + 2b = 120 150 - b = 120 Step 4: Isolate b 150 - 120 = b 30 = b Step 5: Find c using c = 50 - b c = 50 - 30 c = 20 Step 6: Verify the solution Money: 3(20) + 2(30) = 60 + 60 = 120 ✓ Total items: 20 + 30 = 50 ✓ Therefore, Liam should plan to sell 20 boxes of cookies and 30 candy bars.

3. A right triangle is drawn on a coordinate plane with vertices at (0,0), (6,0), and (0,8). A line is drawn from the vertex at (0,8) perpendicular to the hypotenuse, meeting it at point P. Visualize this right triangle and the perpendicular line creating two smaller right triangles within the original triangle. What is the length of the perpendicular segment from (0,8) to point P? Answer: 4.8 Solution: Find the hypotenuse of the original triangle using the Pythagorean theorem. The legs are 6 and 8, so hypotenuse = sqrt(6^2 + 8^2) = sqrt(36 + 64) = sqrt(100) = 10.
   Full step-by-step solution

   Step 1: Find the hypotenuse of the original triangle using the Pythagorean theorem. The legs are 6 and 8, so hypotenuse = sqrt(6^2 + 8^2) = sqrt(36 + 64) = sqrt(100) = 10. Step 2: The perpendicular from the right angle vertex to the hypotenuse creates two smaller triangles that are similar to the original triangle. Step 3: The area of the original triangle can be calculated in two ways. Area = (1/2) × base × height = (1/2) × 6 × 8 = 24 square units. Step 4: The same area can also be expressed as (1/2) × hypotenuse × perpendicular height from right angle vertex. So, 24 = (1/2) × 10 × perpendicular height. Step 5: Solve for the perpendicular height. 24 = 5 × perpendicular height perpendicular height = 24 ÷ 5 = 4.8 Step 6: Therefore, the length of the perpendicular segment from (0,8) to point P is 4.8 units.

4. A science class is growing bacteria in two different petri dishes. Dish A starts with 8,000 bacteria and grows at a rate of 400 bacteria per hour. Dish B starts with 2,000 bacteria and grows at a rate of 800 bacteria per hour. After how many hours will both dishes contain the same number of bacteria? Answer: 15 Solution: Let \( t \) = number of hours after the start. Dish A starts with 8,000 bacteria and grows at 400 bacteria per hour.
   Full step-by-step solution

   Let's solve this step by step. --- **Step 1: Define variables** Let \( t \) = number of hours after the start. Dish A starts with 8,000 bacteria and grows at 400 bacteria per hour. So, number of bacteria in Dish A after \( t \) hours is: \[ A(t) = 8000 + 400t \] Dish B starts with 2,000 bacteria and grows at 800 bacteria per hour. So, number of bacteria in Dish B after \( t \) hours is: \[ B(t) = 2000 + 800t \] --- **Step 2: Set up the equation** We want the time \( t \) when both dishes have the same number of bacteria: \[ 8000 + 400t = 2000 + 800t \] --- **Step 3: Solve for \( t \)** Subtract 2000 from both sides: \[ 8000 - 2000 + 400t = 800t \] \[ 6000 + 400t = 800t \] Subtract 400t from both sides: \[ 6000 = 800t - 400t \] \[ 6000 = 400t \] Divide both sides by 400: \[ t = 6000 / 400 \] \[ t = 15 \] --- **Step 4: Interpret the result** After 15 hours, both dishes will have the same number of bacteria. --- **Step 5: Check the answer** Dish A after 15 hours: \( 8000 + 400 \times 15 = 8000 + 6000 = 14000 \) Dish B after 15 hours: \( 2000 + 800 \times 15 = 2000 + 12000 = 14000 \) They match. --- **Final answer:** 15

5. Charlotte and Isabella are selling cookies for a fundraiser. Charlotte sells boxes of chocolate chip cookies for $9 each, and Isabella sells boxes of oatmeal cookies for $11 each. Together, they sold 48 boxes and earned a total of $468. How many boxes did each person sell? Answer: Charlotte sold 30 boxes, Isabella sold 18 boxes Solution: Let c = number of boxes Charlotte sold, and i = number of boxes Isabella sold. Write the equations. Total boxes: c + i = 48 Total money: 9c + 11i = 468 Solve by elimination.
   Full step-by-step solution

   Step 1: Let c = number of boxes Charlotte sold, and i = number of boxes Isabella sold. Step 2: Write the equations. Total boxes: c + i = 48 Total money: 9c + 11i = 468 Step 3: Solve by elimination. Multiply the first equation by 9: 9c + 9i = 432. Step 4: Subtract from the second equation: (9c + 11i) - (9c + 9i) = 468 - 432, so 2i = 36, i = 18. Step 5: Substitute i = 18 into c + i = 48: c + 18 = 48, so c = 30. Step 6: Check: 30 + 18 = 48 boxes, and 9(30) + 11(18) = 270 + 198 = 468 dollars. Correct. The answer is Charlotte sold 30 boxes and Isabella sold 18 boxes.

6. A rectangular garden is drawn on a coordinate plane with corners at points (2, 1), (8, 1), (8, 5), and (2, 5). A diagonal is drawn from (2, 1) to (8, 5), creating two right triangles. What is the area of one of these right triangles? Answer: 12 Solution: A = (2, 1) B = (8, 1) C = (8, 5) D = (2, 5) A to B is along the bottom side (y = 1) B to C is along the right side (x = 8) C to D is along the top side (y = 5) D to A is along the left side (x = 2) Length AB = 8 - 2 = 6 Width BC = 5 - 1 = 4 So the rectangle's area = length × width = 6 × 4 = 24.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the rectangle and diagonal** The rectangle's corners are: A = (2, 1) B = (8, 1) C = (8, 5) D = (2, 5) Plotting these: A to B is along the bottom side (y = 1) B to C is along the right side (x = 8) C to D is along the top side (y = 5) D to A is along the left side (x = 2) --- **Step 2: Find the rectangle's dimensions** Length AB = 8 - 2 = 6 Width BC = 5 - 1 = 4 So the rectangle's area = length × width = 6 × 4 = 24. --- **Step 3: Diagonal divides rectangle into two equal triangles** The diagonal from A(2, 1) to C(8, 5) splits the rectangle into two congruent right triangles: triangle ABC and triangle ADC. Each triangle's area is half the rectangle's area: Area of one triangle = 24 / 2 = 12. --- **Step 4: Verify by triangle area formula (optional check)** Take triangle ABC: Right angle at B(8, 1) Base AB = 6 Height BC = 4 Area = (1/2) × base × height = (1/2) × 6 × 4 = 12. Same for triangle ADC: Right angle at D(2, 5) Base AD = 4 Height DC = 6 Area = (1/2) × 4 × 6 = 12. --- **Final answer:** 12

7. Ava is designing a community garden and needs to determine how many rectangular plots and circular plots to create. Each rectangular plot requires 8 square meters of space, while each circular plot requires 5 square meters. The total available garden space is 200 square meters. Additionally, Ava wants exactly 30 plots in total to accommodate all the community members who signed up. How many rectangular plots and how many circular plots should Ava create to use exactly all the available space and meet the total plot requirement? Answer: x = 50/3, y = 40/3 Solution: When solving problems with two constraints, we often use a system of equations. One equation typically represents the total quantity of items, while the other represents a measurement like total cost, area, or weight.
   Full step-by-step solution

   When solving problems with two constraints, we often use a system of equations. One equation typically represents the total quantity of items, while the other represents a measurement like total cost, area, or weight. The solution helps find how many of each type meets both conditions simultaneously.