Linear with Rationals

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Answer Key & Explanations

Linear with Rationals · Grade 9 · Worksheet 1

1. 3(2x - 5) + 4(x + 3) = 7x - 2 Answer: x = 1 Solution: When solving linear equations with rational coefficients, first apply the distributive property to remove parentheses. Then combine all like terms on each side of the equation.
   Full step-by-step solution

   When solving linear equations with rational coefficients, first apply the distributive property to remove parentheses. Then combine all like terms on each side of the equation. Finally, use inverse operations to isolate the variable term on one side and the constant terms on the other side.

2. A rectangular prism is drawn on a coordinate plane with vertices at (0,0,0), (x,0,0), (0,2x,0), (0,0,3), (x,2x,0), (x,0,3), (0,2x,3), and (x,2x,3). The volume of the prism is 150 cubic units. Find the value of x. Answer: 5 Solution: Identify the dimensions from the coordinates. The prism extends from x=0 to x=x, so length = x. The prism extends from y=0 to y=2x, so width = 2x.
   Full step-by-step solution

   Step 1: Identify the dimensions from the coordinates. The prism extends from x=0 to x=x, so length = x. The prism extends from y=0 to y=2x, so width = 2x. The prism extends from z=0 to z=3, so height = 3. Step 2: Write the volume formula for a rectangular prism. Volume = length × width × height Step 3: Substitute the known values. 150 = x × (2x) × 3 Step 4: Simplify the equation. 150 = 6x^2 Step 5: Solve for x^2. x^2 = 150 / 6 x^2 = 25 Step 6: Solve for x. x = sqrt(25) x = 5 The answer is 5.

3. A physics class is studying projectile motion. The height h (in meters) of a ball thrown upward is given by the function h(t) = -5t² + 20t + 1.5, where t is time in seconds. At what time does the ball reach its maximum height? Answer: 2 Solution: The height function is h(t) = -5t² + 20t + 1.5 This is a quadratic function in the form at² + bt + c, where a = -5, b = 20, c = 1.5 For a quadratic function, the maximum or minimum occurs at t = -b/(2a) Substitute the values: t = -20/(2×(-5)) = -20/(-10) = 2 The ball reaches its maximum height…
   Full step-by-step solution

   Step 1: The height function is h(t) = -5t² + 20t + 1.5 Step 2: This is a quadratic function in the form at² + bt + c, where a = -5, b = 20, c = 1.5 Step 3: For a quadratic function, the maximum or minimum occurs at t = -b/(2a) Step 4: Substitute the values: t = -20/(2×(-5)) = -20/(-10) = 2 Step 5: The ball reaches its maximum height at t = 2 seconds The answer is 2.

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 65 square meters. What is the width of Liam's garden? Answer: 5 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 the 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. Given area = 65 square meters: \[ (2w + 3) \times w = 65 \] --- **Step 3: Expand and rearrange** \[ 2w^2 + 3w = 65 \] \[ 2w^2 + 3w - 65 = 0 \] --- **Step 4: Solve the quadratic equation** Use the quadratic formula: \( w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) Here \( a = 2 \), \( b = 3 \), \( c = -65 \). First, discriminant: \[ b^2 - 4ac = 3^2 - 4(2)(-65) = 9 + 520 = 529 \] \[ \sqrt{529} = 23 \] So: \[ w = \frac{-3 \pm 23}{2 \times 2} = \frac{-3 \pm 23}{4} \] --- **Step 5: Two possible solutions** \[ w = \frac{-3 + 23}{4} = \frac{20}{4} = 5 \] \[ w = \frac{-3 - 23}{4} = \frac{-26}{4} = -6.5 \] --- **Step 6: Choose the valid solution** Width cannot be negative, so \( w = 5 \) meters. --- **Step 7: Check** Width \( w = 5 \) m Length \( l = 2(5) + 3 = 13 \) m Area = \( 13 \times 5 = 65 \) m² ✔ --- **Final answer:** The width is 5 meters.

5. A right triangle is drawn on a coordinate plane with vertices at (0,0), (x,0), and (0,2x). The hypotenuse has a length of 10√5 units. Find the value of x. Answer: 10 Solution: - A = (0, 0) - B = (x, 0) - C = (0, 2x) - AB is along the x-axis from (0,0) to (x,0), length = x - AC is along the y-axis from (0,0) to (0,2x), length = 2x - BC is the hypotenuse from (x,0) to (0,2x) Distance between B(x, 0) and C(0, 2x): BC = sqrt( (0 - x)^2 + (2x - 0)^2 ) = sqrt( (-x)^2 +…
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the triangle's vertices** Vertices are: - A = (0, 0) - B = (x, 0) - C = (0, 2x) So: - AB is along the x-axis from (0,0) to (x,0), length = x - AC is along the y-axis from (0,0) to (0,2x), length = 2x - BC is the hypotenuse from (x,0) to (0,2x) --- **Step 2: Apply the distance formula for BC** Distance between B(x, 0) and C(0, 2x): BC = sqrt( (0 - x)^2 + (2x - 0)^2 ) = sqrt( (-x)^2 + (2x)^2 ) = sqrt( x^2 + 4x^2 ) = sqrt( 5x^2 ) --- **Step 3: Simplify sqrt(5x^2)** sqrt(5x^2) = sqrt(5) * sqrt(x^2) = sqrt(5) * |x| Since x > 0 (lengths are positive), |x| = x. So BC = x * sqrt(5) --- **Step 4: Set BC equal to given hypotenuse length** We are told BC = 10 * sqrt(5) So: x * sqrt(5) = 10 * sqrt(5) --- **Step 5: Solve for x** Divide both sides by sqrt(5): x = 10 --- **Step 6: Check** If x = 10: - AB = 10 - AC = 20 - BC = sqrt(10^2 + 20^2) = sqrt(100 + 400) = sqrt(500) = sqrt(100*5) = 10*sqrt(5) ✔ --- **Final answer:** x = 10

6. A robotics team is programming a drone's flight path. The drone's altitude follows the function A(t) = (2t² - 18) / (t - 3), where t is time in seconds and A(t) is altitude in meters. At what time does the drone reach ground level? Answer: -3 Solution: Set the altitude function equal to zero: (2t² - 18) / (t - 3) = 0 A rational function equals zero when its numerator equals zero (and denominator doesn't make it undefined): 2t² - 18 = 0 Solve the numerator equation: 2t² - 18 = 0 → 2t² = 18 → t² = 9 → t = 3 or t = -3 Check for restrictions from…
   Full step-by-step solution

   Step 1: Set the altitude function equal to zero: (2t² - 18) / (t - 3) = 0 Step 2: A rational function equals zero when its numerator equals zero (and denominator doesn't make it undefined): 2t² - 18 = 0 Step 3: Solve the numerator equation: 2t² - 18 = 0 → 2t² = 18 → t² = 9 → t = 3 or t = -3 Step 4: Check for restrictions from the denominator: t - 3 ≠ 0 → t ≠ 3 Step 5: Eliminate t = 3 since it makes the denominator zero Step 6: The valid solution is t = -3 The drone reaches ground level at t = -3 seconds.

7. Mere is calculating the cost of a school trip. The total cost C (in dollars) for x students is given by the equation C = (3/4)x + 25. If the total cost for the trip is $85, how many students are going on the trip? Answer: 80 Solution: Write the equation: (3/4)x + 25 = 85 Subtract 25 from both sides: (3/4)x = 60 Multiply both sides by the reciprocal of 3/4, which is 4/3: x = 60 * (4/3) Simplify: x = (60 * 4) / 3 = 240 / 3 = 80 The answer is 80 students.
   Full step-by-step solution

   Step 1: Write the equation: (3/4)x + 25 = 85 Step 2: Subtract 25 from both sides: (3/4)x = 60 Step 3: Multiply both sides by the reciprocal of 3/4, which is 4/3: x = 60 * (4/3) Step 4: Simplify: x = (60 * 4) / 3 = 240 / 3 = 80 The answer is 80 students.

8. Sophia is analyzing the temperature change in a chemical reaction. The temperature T (in degrees Celsius) of the solution is modeled by the equation 0.6t - 1.4 = 3.1, where t represents time in minutes after the reaction begins. At what time does the temperature reach the target value? Answer: 7.5 Solution: Start with the equation: 0.6t - 1.4 = 3.1 Add 1.4 to both sides to isolate the term with t: 0.6t - 1.4 + 1.4 = 3.1 + 1.4 Simplify: 0.6t = 4.5 Divide both sides by 0.6 to solve for t: 0.6t / 0.6 = 4.5 / 0.6 Calculate: t = 7.5 The answer is 7.5.
   Full step-by-step solution

   Step 1: Start with the equation: 0.6t - 1.4 = 3.1 Step 2: Add 1.4 to both sides to isolate the term with t: 0.6t - 1.4 + 1.4 = 3.1 + 1.4 Step 3: Simplify: 0.6t = 4.5 Step 4: Divide both sides by 0.6 to solve for t: 0.6t / 0.6 = 4.5 / 0.6 Step 5: Calculate: t = 7.5 The answer is 7.5.

9. 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.