Infinite Solutions

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Answer Key & Explanations

Infinite Solutions · Grade 8 · Worksheet 3

1. A rectangle is drawn on a coordinate plane with vertices at (2, 1), (8, 1), (8, 5), and (2, 5). A line passes through the midpoints of the top and bottom sides of this rectangle. What is the equation of this line? Answer: x = 5 Solution: Identify the vertices of the rectangle. (2, 1), (8, 1), (8, 5), (2, 5). - (2, 1) is bottom-left.
   Full step-by-step solution

   Let's go step by step. --- **Step 1: Identify the vertices of the rectangle.** The vertices are given as: (2, 1), (8, 1), (8, 5), (2, 5). Plotting them mentally: - (2, 1) is bottom-left. - (8, 1) is bottom-right. - (8, 5) is top-right. - (2, 5) is top-left. So: - Bottom side: from (2, 1) to (8, 1) - Top side: from (2, 5) to (8, 5) --- **Step 2: Find the midpoints of the top and bottom sides.** Midpoint formula: ( (x1 + x2)/2 , (y1 + y2)/2 ) Bottom side midpoint: ( (2 + 8)/2 , (1 + 1)/2 ) = (10/2, 2/2) = (5, 1) Top side midpoint: ( (2 + 8)/2 , (5 + 5)/2 ) = (10/2, 10/2) = (5, 5) --- **Step 3: Identify the line through these midpoints.** Midpoints are (5, 1) and (5, 5). Notice: Both have the same x-coordinate (x = 5), but different y-coordinates. --- **Step 4: Determine the equation of the line.** If two points have the same x-coordinate, the line is vertical. Equation of a vertical line: x = constant. Here, x = 5 for both points, so the line is x = 5. --- **Step 5: Conclusion** The line passing through the midpoints of the top and bottom sides is a vertical line through x = 5. **Final answer: x = 5** ---

2. Emma is volunteering at a community garden where they are planting flower beds. The garden coordinator tells her that the number of marigold plants in one bed, when multiplied by 5 and then increased by 7, is equal to 7 more than 5 times the number of marigold plants. How many marigold plants are in the bed, and what does this tell you about the number of solutions to the equation representing this situation? Answer: infinitely many solutions Solution: Let m represent the number of marigold plants. The equation from the problem is: 5m + 7 = 5m + 7. Subtract 5m from both sides: 5m + 7 - 5m = 5m + 7 - 5m, which simplifies to 7 = 7.
   Full step-by-step solution

   Step 1: Let m represent the number of marigold plants. Step 2: The equation from the problem is: 5m + 7 = 5m + 7. Step 3: Subtract 5m from both sides: 5m + 7 - 5m = 5m + 7 - 5m, which simplifies to 7 = 7. Step 4: The statement 7 = 7 is always true, regardless of the value of m. Step 5: This means the equation is an identity, and there are infinitely many solutions. Any number of marigold plants would satisfy the condition. Answer: Infinitely many solutions.

3. Sophia is helping her aunt set up a lemonade stand. They use a recipe that calls for a certain number of lemons. The recipe says: 'The number of lemons needed, when multiplied by 6 and then increased by 11, is equal to 6 times the number of lemons plus 11.' How many solutions does this equation have for the number of lemons needed? Answer: infinitely many solutions Solution: Let x represent the number of lemons needed. Write the equation based on the recipe: 6x + 11 = 6x + 11. Subtract 6x from both sides: 6x + 11 - 6x = 6x + 11 - 6x → 11 = 11.
   Full step-by-step solution

   Step 1: Let x represent the number of lemons needed. Step 2: Write the equation based on the recipe: 6x + 11 = 6x + 11. Step 3: Subtract 6x from both sides: 6x + 11 - 6x = 6x + 11 - 6x → 11 = 11. Step 4: This simplifies to a true statement (11 = 11), which is always true, regardless of the value of x. Step 5: Therefore, the equation has infinitely many solutions. Any number of lemons would satisfy the recipe statement. The answer is: infinitely many solutions.

4. Emma is designing a rectangular banner for the school science fair. The length of the banner is 2 meters less than three times its width. If the perimeter of the banner is 44 meters, what are the dimensions (width and length) of the banner? Answer: 6 Solution: Let the width be w meters. The length is 2 meters less than three times the width, so length = 3w - 2. The perimeter of a rectangle is given by P = 2(length + width).
   Full step-by-step solution

   Step 1: Let the width be w meters. Step 2: The length is 2 meters less than three times the width, so length = 3w - 2. Step 3: The perimeter of a rectangle is given by P = 2(length + width). Step 4: Substitute the known perimeter and expressions: 44 = 2((3w - 2) + w). Step 5: Simplify inside the parentheses: 44 = 2(4w - 2). Step 6: Distribute the 2: 44 = 8w - 4. Step 7: Add 4 to both sides: 48 = 8w. Step 8: Divide both sides by 8: w = 6. Step 9: The width is 6 meters. The length is 3(6) - 2 = 18 - 2 = 16 meters. The dimensions are width = 6 m and length = 16 m. The question asks for the width, so the answer is 6.

5. A triangle is drawn on a coordinate plane with vertices at A(2, 3), B(8, 3), and C(5, 7). A line is drawn from vertex C to the midpoint of side AB. What are the coordinates of this midpoint? Answer: (5, 3) Solution: Identify the endpoints of side AB: A(2, 3) and B(8, 3) Find the x-coordinate of the midpoint: (2 + 8) ÷ 2 = 10 ÷ 2 = 5 Find the y-coordinate of the midpoint: (3 + 3) ÷ 2 = 6 ÷ 2 = 3 Combine the coordinates: (5, 3) The midpoint of side AB is (5, 3).
   Full step-by-step solution

   Step 1: Identify the endpoints of side AB: A(2, 3) and B(8, 3) Step 2: Find the x-coordinate of the midpoint: (2 + 8) ÷ 2 = 10 ÷ 2 = 5 Step 3: Find the y-coordinate of the midpoint: (3 + 3) ÷ 2 = 6 ÷ 2 = 3 Step 4: Combine the coordinates: (5, 3) The midpoint of side AB is (5, 3).

6. A rectangle is drawn on a coordinate plane with vertices at (2, 1), (8, 1), (8, 5), and (2, 5). A line passes through the midpoints of the two longer sides of the rectangle. What is the equation of this line in slope-intercept form? Answer: y = 3 Solution: Horizontal lines have a slope of zero and maintain a constant y-value across all x-coordinates. The equation of such lines is always in the form y = constant, where the constant represents the y-coordinate of every point on the line.
7. Liam is designing a rectangular garden with a perimeter of 36 meters. The length of the garden is 4 meters more than twice its width. Write an equation in terms of width w that represents this situation, then find the dimensions of the garden. Answer: width = 4 meters, length = 12 meters Solution: When solving geometry problems with relationships between dimensions, you can use variables to represent unknown quantities.
   Full step-by-step solution

   When solving geometry problems with relationships between dimensions, you can use variables to represent unknown quantities. The perimeter formula for rectangles helps connect these dimensions, and setting up equations based on given relationships allows you to find specific measurements.

8. A triangle is drawn on a coordinate plane with vertices at A(1, 2), B(5, 6), and C(9, 2). A line is drawn from vertex A to the midpoint of side BC. What are the coordinates of this midpoint? Answer: (7, 4) Solution: Identify the endpoints of side BC. B is at (5, 6) and C is at (9, 2). To find the midpoint, average the x-coordinates: (5 + 9) / 2 = 14 / 2 = 7.
   Full step-by-step solution

   Step 1: Identify the endpoints of side BC. B is at (5, 6) and C is at (9, 2). Step 2: To find the midpoint, average the x-coordinates: (5 + 9) / 2 = 14 / 2 = 7. Step 3: Average the y-coordinates: (6 + 2) / 2 = 8 / 2 = 4. Step 4: The midpoint of BC is at (7, 4). The answer is (7, 4).