Linear with Rationals

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Answer Key & Explanations

Linear with Rationals · Grade 9 · Worksheet 3

1. A robotics team is programming a robot's movement path. The robot's position function is given by P(t) = (2t² - 8t + 6)/(t - 1), where t represents time in seconds. At what time does the robot reach its starting position (position = 0)? Answer: 3 Solution: Set the position function equal to zero: (2t² - 8t + 6)/(t - 1) = 0 A fraction equals zero when its numerator equals zero (and denominator is not zero) Set numerator equal to zero: 2t² - 8t + 6 = 0 Divide by 2: t² - 4t + 3 = 0 Factor: (t - 1)(t - 3) = 0 Solve: t = 1 or t = 3 Check denominator: t…
   Full step-by-step solution

   Step 1: Set the position function equal to zero: (2t² - 8t + 6)/(t - 1) = 0 Step 2: A fraction equals zero when its numerator equals zero (and denominator is not zero) Step 3: Set numerator equal to zero: 2t² - 8t + 6 = 0 Step 4: Divide by 2: t² - 4t + 3 = 0 Step 5: Factor: (t - 1)(t - 3) = 0 Step 6: Solve: t = 1 or t = 3 Step 7: Check denominator: t - 1 = 0 when t = 1, so t = 1 is not allowed Step 8: Therefore, the only valid solution is t = 3 The answer is 3.

2. A robotics team is programming a robot to follow a parabolic path. The robot's height above ground is modeled by the function h(t) = -2t² + 12t + 8, where h is height in meters and t is time in seconds. At what time will the robot reach its maximum height? Answer: 3 Solution: The function h(t) = -2t² + 12t + 8 is a quadratic function in the form at² + bt + c, where a = -2, b = 12, and c = 8. For a quadratic function, the maximum or minimum occurs at t = -b/(2a).
   Full step-by-step solution

   Step 1: The function h(t) = -2t² + 12t + 8 is a quadratic function in the form at² + bt + c, where a = -2, b = 12, and c = 8. Step 2: For a quadratic function, the maximum or minimum occurs at t = -b/(2a). Step 3: Substitute the values: t = -12/(2 × -2) = -12/(-4) = 3. Step 4: The robot reaches its maximum height at t = 3 seconds. The answer is 3.

3. (5x - 8)/9 - (3x + 11)/7 = 2/3 Answer: x = 17 Solution: Identify the denominators: 9, 7, and 3. The least common denominator is 63.
   Full step-by-step solution

   Step 1: Identify the denominators: 9, 7, and 3. The least common denominator is 63. Step 2: Multiply both sides of the equation by 63: 63 * [(5x - 8)/9 - (3x + 11)/7] = 63 * (2/3) Step 3: Distribute the 63 to each term on the left: 63 * (5x - 8)/9 - 63 * (3x + 11)/7 = 42 Step 4: Simplify each fraction: 7(5x - 8) - 9(3x + 11) = 42 Step 5: Distribute the coefficients: 35x - 56 - 27x - 99 = 42 Step 6: Combine like terms: (35x - 27x) + (-56 - 99) = 42 8x - 155 = 42 Step 7: Add 155 to both sides: 8x = 197 Step 8: Divide both sides by 8: x = 197/8 x = 24.625 The answer is x = 24.625.

4. Liam is designing a rectangular garden with a length that is 3 meters more than twice its width. The area of the garden must be exactly 65 square meters. What are the dimensions of Liam's garden? Answer: width = 5 m, length = 13 m Solution: Let the width of the garden be \( w \) meters. The length is 3 meters more than twice the width, so: length \( l = 2w + 3 \). Area of a rectangle = length × width.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Define variables** Let the width of the garden be \( w \) meters. The length is 3 meters more than twice the width, so: length \( l = 2w + 3 \). --- **Step 2: Write the area equation** Area of a rectangle = length × width. Area = \( l \times w = (2w + 3) \times w \). Given area = 65 m², so: \[ w(2w + 3) = 65 \] --- **Step 3: Expand and rearrange** \[ 2w^2 + 3w = 65 \] \[ 2w^2 + 3w - 65 = 0 \] --- **Step 4: Solve the quadratic equation** We solve \( 2w^2 + 3w - 65 = 0 \) using the quadratic formula: \[ w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \( a = 2 \), \( b = 3 \), \( c = -65 \). First, discriminant \( D = b^2 - 4ac \): \[ D = 3^2 - 4(2)(-65) = 9 + 520 = 529 \] \[ \sqrt{D} = \sqrt{529} = 23 \] So: \[ w = \frac{-3 \pm 23}{2 \times 2} = \frac{-3 \pm 23}{4} \] --- **Step 5: Two possible solutions** Case 1: \( w = \frac{-3 + 23}{4} = \frac{20}{4} = 5 \) Case 2: \( w = \frac{-3 - 23}{4} = \frac{-26}{4} = -6.5 \) Width cannot be negative, so \( w = 5 \) m. --- **Step 6: Find length** \[ l = 2w + 3 = 2(5) + 3 = 10 + 3 = 13 \ \text{m} \] --- **Step 7: Check area** Area = \( 5 \times 13 = 65 \) m², correct. --- **Final answer:** width = 5 m, length = 13 m

5. Olivia draws a right triangle on a coordinate plane with vertices at (0,0), (7x,0), and (0, (3/5)x). The area of the triangle is 21 square units. Find the value of x. Answer: sqrt(10) Solution: Identify the legs from the coordinates. The base is along the x-axis from (0,0) to (7x,0), so base = 7x. The height is along the y-axis from (0,0) to (0, (3/5)x), so height = (3/5)x.
   Full step-by-step solution

   Step 1: Identify the legs from the coordinates. The base is along the x-axis from (0,0) to (7x,0), so base = 7x. The height is along the y-axis from (0,0) to (0, (3/5)x), so height = (3/5)x. Step 2: Use the area formula for a triangle: Area = (1/2) * base * height. Step 3: Substitute the given area and the expressions: 21 = (1/2) * (7x) * ((3/5)x). Step 4: Simplify the right side: (1/2) * 7x * (3/5)x = (1/2) * 7 * (3/5) * x * x = (1/2) * (21/5) * x^2 = (21/10)x^2. Step 5: The equation becomes: 21 = (21/10)x^2. Step 6: Multiply both sides by 10: 210 = 21x^2. Step 7: Divide both sides by 21: 10 = x^2. Step 8: Take the square root of both sides (x must be positive since it represents length): x = sqrt(10). The answer is sqrt(10).

6. A robotics team is programming a drone to follow a parabolic path described by the equation h(t) = -5t² + 40t, where h represents height in meters and t represents time in seconds. The team wants to know when the drone will return to its launch height of 0 meters. Solve the equation -5t² + 40t = 0 to find the time when the drone returns to ground level. Answer: 8 Solution: Start with the equation -5t² + 40t = 0 Factor out the greatest common factor, which is -5t: -5t(t - 8) = 0 Set each factor equal to zero: -5t = 0 or t - 8 = 0 Solve the first equation: -5t = 0 → t = 0 Solve the second equation: t - 8 = 0 → t = 8 Interpret the solutions: t = 0 represents the…
   Full step-by-step solution

   Step 1: Start with the equation -5t² + 40t = 0 Step 2: Factor out the greatest common factor, which is -5t: -5t(t - 8) = 0 Step 3: Set each factor equal to zero: -5t = 0 or t - 8 = 0 Step 4: Solve the first equation: -5t = 0 → t = 0 Step 5: Solve the second equation: t - 8 = 0 → t = 8 Step 6: Interpret the solutions: t = 0 represents the launch time, and t = 8 represents when the drone returns to ground level. The answer is 8 seconds.

7. Mason is calculating the dimensions of a rectangular plot of land. The length of the plot is 3.75 meters more than twice its width. The perimeter of the plot is 52.5 meters. What is the width of the plot in meters? Answer: 7.5 Solution: Let w represent the width in meters. The length is 3.75 more than twice the width, so length = 2w + 3.75. The perimeter of a rectangle is P = 2(length) + 2(width).
   Full step-by-step solution

   Step 1: Let w represent the width in meters. Step 2: The length is 3.75 more than twice the width, so length = 2w + 3.75. Step 3: The perimeter of a rectangle is P = 2(length) + 2(width). Step 4: Substitute: 52.5 = 2(2w + 3.75) + 2w. Step 5: Distribute: 52.5 = 4w + 7.5 + 2w. Step 6: Combine like terms: 52.5 = 6w + 7.5. Step 7: Subtract 7.5 from both sides: 52.5 - 7.5 = 6w, so 45 = 6w. Step 8: Divide both sides by 6: w = 45 / 6 = 7.5. The width of the plot is 7.5 meters.

8. Jordan is filling a rectangular garden bed that has an area of 36 square meters. The length of the bed is 3 meters. The width, w, is the unknown. The area formula is length × width = area. Write and solve an equation to find the width of the garden bed. Answer: 12 Solution: Use the area formula: length × width = area. Substitute known values: 3 × w = 36. Solve for w by dividing both sides by 3: w = 36 ÷ 3.
   Full step-by-step solution

   Step 1: Use the area formula: length × width = area. Step 2: Substitute known values: 3 × w = 36. Step 3: Solve for w by dividing both sides by 3: w = 36 ÷ 3. Step 4: Calculate: w = 12. The width of the garden bed is 12 meters.