Create Equations

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Answer Key & Explanations

Create Equations · Grade 9 · Worksheet 1

1. Emma is designing a rectangular banner. The length is (2x + 5) cm and the width is x cm. If the perimeter is 150 cm, write an equation to find x. Answer: 2(2x + 5 + x) = 150 Solution: The formula for the perimeter of a rectangle is P = 2(length + width). Substitute the given expressions: length = (2x + 5) cm, width = x cm, and perimeter = 150 cm. This gives us the equation: 2((2x + 5) + x) = 150.
   Full step-by-step solution

   Step 1: The formula for the perimeter of a rectangle is P = 2(length + width). Step 2: Substitute the given expressions: length = (2x + 5) cm, width = x cm, and perimeter = 150 cm. Step 3: This gives us the equation: 2((2x + 5) + x) = 150. Step 4: Simplify inside the parentheses: 2(3x + 5) = 150. The equation to find x is 2(3x + 5) = 150.

2. Isabella is designing a rectangular garden with a perimeter of 68 feet. The length is 4x + 5 feet and the width is 2x - 3 feet. Write an equation to find x. Answer: 2(4x + 5) + 2(2x - 3) = 68 Solution: The formula for the perimeter of a rectangle is P = 2L + 2W, where L is the length and W is the width. Substitute the given expressions: L = 4x + 5 and W = 2x - 3, and the perimeter P = 68.
   Full step-by-step solution

   Step 1: The formula for the perimeter of a rectangle is P = 2L + 2W, where L is the length and W is the width. Step 2: Substitute the given expressions: L = 4x + 5 and W = 2x - 3, and the perimeter P = 68. Step 3: This gives the equation: 2(4x + 5) + 2(2x - 3) = 68. Step 4: The equation is now complete. The answer is 2(4x + 5) + 2(2x - 3) = 68.

3. Tane is building a rectangular enclosure for his sheep. The length of the enclosure is 7 meters more than three times the width. If the perimeter of the enclosure is 174 meters, write an equation to represent this situation and solve for the width of the enclosure. Answer: 20 Solution: Let w represent the width in meters. The length is 7 meters more than three times the width, so length = 3w + 7. The perimeter of a rectangle is given by P = 2(length + width).
   Full step-by-step solution

   Step 1: Let w represent the width in meters. Step 2: The length is 7 meters more than three times the width, so length = 3w + 7. Step 3: The perimeter of a rectangle is given by P = 2(length + width). Step 4: Substitute the known perimeter and expressions: 174 = 2((3w + 7) + w). Step 5: Simplify inside the parentheses: 3w + 7 + w = 4w + 7. Step 6: The equation becomes 174 = 2(4w + 7). Step 7: Divide both sides by 2: 87 = 4w + 7. Step 8: Subtract 7 from both sides: 80 = 4w. Step 9: Divide both sides by 4: w = 20. Step 10: The width of the enclosure is 20 meters. The answer is 20.

4. 2x² - 8x + 8 = 0 Answer: 2 Solution: Factor out the common factor of 2 from all terms: 2(x² - 4x + 4) = 0 Divide both sides by 2: x² - 4x + 4 = 0 Recognize this as a perfect square trinomial: (x - 2)² = 0 Take the square root of both sides: x - 2 = 0 Add 2 to both sides: x = 2 The answer is 2.
   Full step-by-step solution

   Step 1: Factor out the common factor of 2 from all terms: 2(x² - 4x + 4) = 0 Step 2: Divide both sides by 2: x² - 4x + 4 = 0 Step 3: Recognize this as a perfect square trinomial: (x - 2)² = 0 Step 4: Take the square root of both sides: x - 2 = 0 Step 5: Add 2 to both sides: x = 2 The answer is 2.

5. Tane is designing a rectangular banner. The length of the banner is 11 cm more than the width. The area of the banner is 476 square centimeters. Write an equation to find the width of the banner. Answer: w(w + 11) = 476 Solution: Let w represent the width of the banner in centimeters. The length is 11 cm more than the width, so length = w + 11. The area of a rectangle is given by A = length × width.
   Full step-by-step solution

   Step 1: Let w represent the width of the banner in centimeters. Step 2: The length is 11 cm more than the width, so length = w + 11. Step 3: The area of a rectangle is given by A = length × width. Step 4: Substitute the expressions: (w + 11) × w = 476. Step 5: The equation is w(w + 11) = 476.

6. Tane is designing a rectangular garden. The length of the garden is 12 meters more than three times the width. The area of the garden is 315 square meters. Write an equation to find the width of the garden. Answer: w(3w + 12) = 315 Solution: Let w represent the width of the garden in meters. The length is 12 meters more than three times the width, so length = 3w + 12. The area of a rectangle is given by A = length × width.
   Full step-by-step solution

   Step 1: Let w represent the width of the garden in meters. Step 2: The length is 12 meters more than three times the width, so length = 3w + 12. Step 3: The area of a rectangle is given by A = length × width. Step 4: Substitute the expressions: (3w + 12) × w = 315. Step 5: The equation is w(3w + 12) = 315.

7. Aroha is designing a rectangular banner for a school event. The length of the banner is 9 cm more than three times the width. The area of the banner is 210 square centimeters. Write an equation to find the width of the banner. Answer: w(3w + 9) = 210 Solution: Let w represent the width of the banner in centimeters. The length is 9 cm more than three times the width, so length = 3w + 9. The area of a rectangle is given by A = length × width.
   Full step-by-step solution

   Step 1: Let w represent the width of the banner in centimeters. Step 2: The length is 9 cm more than three times the width, so length = 3w + 9. Step 3: The area of a rectangle is given by A = length × width. Step 4: Substitute the expressions: (3w + 9) × w = 210. Step 5: The equation is w(3w + 9) = 210.

8. Isabella is designing a rectangular garden with a perimeter of 72 meters. The length is 7 meters more than twice the width. Write an equation to find the width. Answer: 2w + 2(2w + 7) = 72 Solution: The length is 7 meters more than twice the width, so length = 2w + 7 The perimeter formula for a rectangle is P = 2(length) + 2(width) Substitute the expressions: 2(2w + 7) + 2(w) = 72 The equation is 2w + 2(2w + 7) = 72
   Full step-by-step solution

   Step 1: Let w represent the width of the garden Step 2: The length is 7 meters more than twice the width, so length = 2w + 7 Step 3: The perimeter formula for a rectangle is P = 2(length) + 2(width) Step 4: Substitute the expressions: 2(2w + 7) + 2(w) = 72 Step 5: The equation is 2w + 2(2w + 7) = 72

9. A right triangle is inscribed in a circle such that the hypotenuse is the diameter of the circle. The legs of the triangle have lengths of 6 cm and 8 cm. What is the area of the circle? (Use π = 3.14) Answer: 78.5 Solution: A right triangle is inscribed in a circle with its hypotenuse as the diameter. The legs are 6 cm and 8 cm. We need to find the area of the circle.
   Full step-by-step solution

   Step 1: Understand the problem. A right triangle is inscribed in a circle with its hypotenuse as the diameter. The legs are 6 cm and 8 cm. We need to find the area of the circle. Step 2: Find the hypotenuse. Since it's a right triangle, use the Pythagorean theorem: Hypotenuse^2 = (leg1)^2 + (leg2)^2 Hypotenuse^2 = 6^2 + 8^2 Hypotenuse^2 = 36 + 64 Hypotenuse^2 = 100 Hypotenuse = sqrt(100) = 10 cm. Step 3: Relate the hypotenuse to the circle. The hypotenuse is the diameter of the circle. So: Diameter (d) = 10 cm. Step 4: Find the radius. Radius (r) = Diameter / 2 = 10 / 2 = 5 cm. Step 5: Find the area of the circle. Area = π * r^2 Area = 3.14 * (5)^2 Area = 3.14 * 25 Area = 78.5 cm^2. Step 6: Final answer. The area of the circle is 78.5 square centimeters.