Linear Equations

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Answer Key & Explanations

Linear Equations · Grade 8 · Worksheet 2

1. Liam is designing a rectangular garden with a length that is 5 meters more than twice its width. The perimeter of the garden is 46 meters. What are the dimensions of Liam's garden? Answer: width = 6 meters, length = 17 meters Solution: Let the width of the garden be \( w \) meters. The length is 5 meters more than twice the width, so: length \( l = 2w + 5 \). \( P = 2 \times (\text{length} + \text{width}) \).
   Full step-by-step solution

   Let's solve step by step. --- **Step 1: Define variables** Let the width of the garden be \( w \) meters. The length is 5 meters more than twice the width, so: length \( l = 2w + 5 \). --- **Step 2: Write the perimeter formula** The perimeter \( P \) of a rectangle is: \( P = 2 \times (\text{length} + \text{width}) \). We are told \( P = 46 \), so: \( 2 \times (l + w) = 46 \). --- **Step 3: Substitute the expression for length** Substitute \( l = 2w + 5 \) into the perimeter equation: \( 2 \times ( (2w + 5) + w ) = 46 \). --- **Step 4: Simplify inside the parentheses** \( (2w + 5) + w = 3w + 5 \). So: \( 2 \times (3w + 5) = 46 \). --- **Step 5: Solve for \( w \)** Divide both sides by 2: \( 3w + 5 = 23 \). Subtract 5 from both sides: \( 3w = 18 \). Divide by 3: \( w = 6 \). --- **Step 6: Find the length** \( l = 2w + 5 = 2 \times 6 + 5 = 12 + 5 = 17 \). --- **Step 7: State the answer** Width = 6 meters, Length = 17 meters. --- **Final check:** Perimeter = \( 2 \times (17 + 6) = 2 \times 23 = 46 \), correct. Length is 5 more than twice the width: \( 2 \times 6 + 5 = 17 \), correct.

2. 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) to a point on the hypotenuse that divides the triangle into two regions of equal area. What is the equation of this line in slope-intercept form? Answer: y = -4/3x + 8 Solution: When a line divides a triangle into two regions of equal area, it creates two smaller triangles that share certain properties with the original triangle.
   Full step-by-step solution

   When a line divides a triangle into two regions of equal area, it creates two smaller triangles that share certain properties with the original triangle. The area formula for triangles and the concept of similar triangles can help determine the exact point where the division occurs. Once you find this point, you can use the coordinates of both endpoints to determine the line's equation.

3. A rectangular garden is drawn on a coordinate plane with vertices at (2, 1), (8, 1), (8, 5), and (2, 5). If you need to place a fence along the entire perimeter of this garden, how many units of fencing are required? Answer: 20 Solution: Identify the vertices of the rectangle in order. Vertices: (2, 1), (8, 1), (8, 5), (2, 5). Points (2, 1) and (8, 1) have the same y-coordinate, so they form a horizontal side.
   Full step-by-step solution

   Let's solve this step by step. Step 1: Identify the vertices of the rectangle in order. Vertices: (2, 1), (8, 1), (8, 5), (2, 5). Step 2: Understand the layout. Points (2, 1) and (8, 1) have the same y-coordinate, so they form a horizontal side. Points (8, 1) and (8, 5) have the same x-coordinate, so they form a vertical side. Similarly, (8, 5) and (2, 5) are horizontal, and (2, 5) and (2, 1) are vertical. Step 3: Find the length of the horizontal side. From (2, 1) to (8, 1): difference in x = 8 - 2 = 6 units. So length = 6. Step 4: Find the length of the vertical side. From (2, 1) to (2, 5): difference in y = 5 - 1 = 4 units. So width = 4. Step 5: Perimeter formula for a rectangle: P = 2 * (length + width). P = 2 * (6 + 4) P = 2 * 10 P = 20 Step 6: Conclusion. The total fencing needed is 20 units. Answer: 20

4. A rectangular prism is drawn with dimensions: length = 2x + 3 cm, width = x - 1 cm, and height = 5 cm. The volume of this prism is 150 cubic centimeters. Find the value of x that satisfies this condition. Answer: 3 Solution: Write the volume formula for a rectangular prism: Volume = length × width × height Substitute the given expressions: (2x + 3)(x - 1)(5) = 150 Divide both sides by 5: (2x + 3)(x - 1) = 30 Expand the left side: 2x(x - 1) + 3(x - 1) = 2x² - 2x + 3x - 3 = 2x² + x - 3 Set up the equation: 2x² + x - 3…
   Full step-by-step solution

   Step 1: Write the volume formula for a rectangular prism: Volume = length × width × height Step 2: Substitute the given expressions: (2x + 3)(x - 1)(5) = 150 Step 3: Divide both sides by 5: (2x + 3)(x - 1) = 30 Step 4: Expand the left side: 2x(x - 1) + 3(x - 1) = 2x² - 2x + 3x - 3 = 2x² + x - 3 Step 5: Set up the equation: 2x² + x - 3 = 30 Step 6: Subtract 30 from both sides: 2x² + x - 33 = 0 Step 7: Factor the quadratic equation: (2x + 11)(x - 3) = 0 Step 8: Solve for x: 2x + 11 = 0 gives x = -11/2, or x - 3 = 0 gives x = 3 Step 9: Since dimensions cannot be negative, we discard x = -11/2 Step 10: The valid solution is x = 3 The answer is 3.

5. Emma is planning a road trip and needs to calculate her fuel costs. Her car's fuel efficiency is 28 miles per gallon, and she plans to drive 350 miles. If gasoline costs $3.25 per gallon, how much will Emma spend on gasoline for her trip? Round your answer to the nearest cent. Answer: 40.63 Solution: Distance = 350 miles Fuel efficiency = 28 miles per gallon Gallons needed = 350 ÷ 28 = 12.5 gallons Cost per gallon = $3.25 Total cost = 12.5 × 3.25 = $40.625 $40.625 rounds to $40.63 The answer is $40.63.
   Full step-by-step solution

   Step 1: Calculate how many gallons of gasoline Emma needs Distance = 350 miles Fuel efficiency = 28 miles per gallon Gallons needed = 350 ÷ 28 = 12.5 gallons Step 2: Calculate the total cost Cost per gallon = $3.25 Total cost = 12.5 × 3.25 = $40.625 Step 3: Round to the nearest cent $40.625 rounds to $40.63 The answer is $40.63.

6. Liam is designing a rectangular garden. The length of the garden is 5 feet more than twice its width. If the perimeter of the garden is 82 feet, what is the width of the garden in feet? Answer: 12 Solution: Let \( w \) = width of the garden in feet.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Define variables** Let \( w \) = width of the garden in feet. The length \( l \) is 5 feet more than twice the width, so: \( l = 2w + 5 \) --- **Step 2: Write the perimeter formula** Perimeter of a rectangle: \( P = 2l + 2w \) Given \( P = 82 \), so: \( 2l + 2w = 82 \) --- **Step 3: Substitute \( l \) into the perimeter equation** Replace \( l \) with \( 2w + 5 \): \( 2(2w + 5) + 2w = 82 \) --- **Step 4: Simplify and solve for \( w \)** First, distribute: \( 4w + 10 + 2w = 82 \) Combine like terms: \( 6w + 10 = 82 \) Subtract 10 from both sides: \( 6w = 72 \) Divide both sides by 6: \( w = 12 \) --- **Step 5: Interpret the result** The width \( w = 12 \) feet. Check: length \( l = 2(12) + 5 = 29 \) feet. Perimeter = \( 2(29) + 2(12) = 58 + 24 = 82 \) feet. ✔ --- **Final answer:** The width is **12 feet**.

7. Liam is designing a rectangular garden with a length that is 5 meters more than twice its width. The perimeter of the garden is 46 meters. What is the width of the garden? Answer: 6 Solution: Let \( w \) = width of the garden (in meters). Length \( l \) = 5 meters more than twice the width, so: \( l = 2w + 5 \). \( P = 2 \times (l + w) \).
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Define the variables** Let \( w \) = width of the garden (in meters). Length \( l \) = 5 meters more than twice the width, so: \( l = 2w + 5 \). --- **Step 2: Write the perimeter formula** Perimeter of a rectangle: \( P = 2 \times (l + w) \). We are told \( P = 46 \), so: \( 2 \times (l + w) = 46 \). --- **Step 3: Substitute \( l \) into the perimeter equation** \( l + w = (2w + 5) + w = 3w + 5 \). So: \( 2 \times (3w + 5) = 46 \). --- **Step 4: Solve for \( w \)** First, divide both sides by 2: \( 3w + 5 = 23 \). Subtract 5 from both sides: \( 3w = 18 \). Divide by 3: \( w = 6 \). --- **Step 5: Interpret the result** The width is 6 meters. --- **Final answer:** 6

8. 2(3x - 5) + 4 = 3(x + 2) - 1 = ? Answer: x = 3 Solution: 2(3x - 5) + 4 = 3(x + 2) - 1 6x - 10 + 4 = 3x + 6 - 1 6x - 6 = 3x + 5 Get all x terms on one side by subtracting 3x from both sides 6x - 6 - 3x = 3x + 5 - 3x 3x - 6 = 5 Isolate the x term by adding 6 to both sides 3x - 6 + 6 = 5 + 6 3x = 11 Solve for x by dividing both sides by 3 3x/3 = 11/3 x =…
   Full step-by-step solution

   Step 1: Distribute on both sides 2(3x - 5) + 4 = 3(x + 2) - 1 6x - 10 + 4 = 3x + 6 - 1 Step 2: Combine like terms on each side 6x - 6 = 3x + 5 Step 3: Get all x terms on one side by subtracting 3x from both sides 6x - 6 - 3x = 3x + 5 - 3x 3x - 6 = 5 Step 4: Isolate the x term by adding 6 to both sides 3x - 6 + 6 = 5 + 6 3x = 11 Step 5: Solve for x by dividing both sides by 3 3x/3 = 11/3 x = 11/3 Step 6: Check the solution 2(3*(11/3) - 5) + 4 = 3((11/3) + 2) - 1 2(11 - 5) + 4 = 3(11/3 + 6/3) - 1 2(6) + 4 = 3(17/3) - 1 12 + 4 = 17 - 1 16 = 16 ✓ The answer is x = 11/3.