Linear with Rationals

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Linear with Rationals · Grade 9 · Worksheet 1

1. A rectangular garden has a length that is 3/2 times its width. If the perimeter of the garden is 60 meters, what is the area of the garden in square meters? Answer: 216 Solution: Let w = width of the garden The length is 3/2 times the width, so length = (3/2)w The perimeter formula is P = 2(length + width) Substitute the known perimeter: 60 = 2((3/2)w + w) Simplify inside the parentheses: 60 = 2((3/2)w + (2/2)w) = 2((5/2)w) Multiply: 60 = (10/2)w = 5w Solve for w: w = 60…
   Full step-by-step solution

   Step 1: Let w = width of the garden Step 2: The length is 3/2 times the width, so length = (3/2)w Step 3: The perimeter formula is P = 2(length + width) Step 4: Substitute the known perimeter: 60 = 2((3/2)w + w) Step 5: Simplify inside the parentheses: 60 = 2((3/2)w + (2/2)w) = 2((5/2)w) Step 6: Multiply: 60 = (10/2)w = 5w Step 7: Solve for w: w = 60 ÷ 5 = 12 meters Step 8: Find the length: length = (3/2) × 12 = 18 meters Step 9: Calculate the area: area = length × width = 18 × 12 = 216 The answer is 216.

2. 2x + 5 = 3(x - 1) + 7 Answer: x = 1 Solution: 2x + 5 = 3(x - 1) + 7 3(x - 1) means 3 times x and 3 times -1. So 3(x - 1) = 3x - 3. 2x + 5 = 3x - 3 + 7 Combine -3 + 7 = 4.
   Full step-by-step solution

   Let's solve the equation step-by-step. We start with: 2x + 5 = 3(x - 1) + 7 **Step 1: Expand the right-hand side** 3(x - 1) means 3 times x and 3 times -1. So 3(x - 1) = 3x - 3. Now the equation becomes: 2x + 5 = 3x - 3 + 7 **Step 2: Simplify the right-hand side** Combine -3 + 7 = 4. So: 2x + 5 = 3x + 4 **Step 3: Move all terms with x to one side** Subtract 2x from both sides: 2x + 5 - 2x = 3x + 4 - 2x This simplifies to: 5 = x + 4 **Step 4: Isolate x** Subtract 4 from both sides: 5 - 4 = x + 4 - 4 1 = x So the solution is: x = 1 **Step 5: Check the solution** Plug x = 1 into the original equation: Left side: 2(1) + 5 = 2 + 5 = 7 Right side: 3(1 - 1) + 7 = 3(0) + 7 = 0 + 7 = 7 Both sides are equal, so x = 1 is correct. Final answer: x = 1

3. The sum of two numbers is 24. One number is 3/4 of the other number. What is the larger number? Answer: 13.7142857143 Solution: Write the equations from the problem. The sum of the two numbers is 24: x + y = 24 One number is 3/4 of the other number.
   Full step-by-step solution

   Let's call the two numbers x and y, where x is the larger number. Step 1: Write the equations from the problem. The sum of the two numbers is 24: x + y = 24 One number is 3/4 of the other number. Since x is the larger number, the smaller number y must be 3/4 of x: y = (3/4) * x Step 2: Substitute the second equation into the first equation. Replace y in the first equation with (3/4)x: x + (3/4)x = 24 Step 3: Combine like terms. x is the same as (4/4)x, so: (4/4)x + (3/4)x = 24 (7/4)x = 24 Step 4: Solve for x. Multiply both sides by 4/7 to isolate x: x = 24 * (4/7) Step 5: Calculate the value. x = (24 * 4) / 7 x = 96 / 7 Step 6: Perform the division. 96 divided by 7 is 13.7142857143... (the decimal repeats 142857). So the larger number is 13.7142857143. Step 7: Verify the result. If x = 96/7, then y = (3/4) * (96/7) = (288/4) / 7 = 72/7 = 10.2857142857... Now check the sum: 96/7 + 72/7 = 168/7 = 24. This confirms our answer is correct.

4. A chemical engineer is designing a reactor where the concentration of a reactant decreases over time according to the function C(t) = (2t² - 18) / (t - 3), where t represents time in hours and t > 3. At what time does the concentration reach zero? Answer: 3 Solution: Step 1: Set the concentration function equal to zero: C(t) = (2t² - 18)/(t - 3) = 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 - 18 = 0 Step 4: Add 18 to both sides: 2t² = 18 Step 5: Divide both sides by 2:…
   Full step-by-step solution

   Step 1: Set the concentration function equal to zero: C(t) = (2t² - 18)/(t - 3) = 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 - 18 = 0 Step 4: Add 18 to both sides: 2t² = 18 Step 5: Divide both sides by 2: t² = 9 Step 6: Take square root of both sides: t = 3 or t = -3 Step 7: Check domain restrictions: t > 3, so t = -3 is invalid Step 8: Verify t = 3: Denominator would be 3 - 3 = 0, which is undefined Step 9: The concentration reaches zero when t = 3 hours, but this is the limit point where the function is undefined The answer is 3.

5. Kaia draws a right triangle on a coordinate plane with vertices at (0,0), (3x,0), and (0, (1/3)x). The area of the triangle is 27 square units. Find the value of x. Answer: sqrt(54) or 3*sqrt(6) or approximately 7.35 Solution: Identify the base and height from the coordinates. The base is along the x-axis from (0,0) to (3x,0), so base length = 3x. The height is along the y-axis from (0,0) to (0, (1/3)x), so height length = (1/3)x.
   Full step-by-step solution

   Step 1: Identify the base and height from the coordinates. The base is along the x-axis from (0,0) to (3x,0), so base length = 3x. The height is along the y-axis from (0,0) to (0, (1/3)x), so height length = (1/3)x. Step 2: Use the area formula for a triangle: Area = (1/2) * base * height Step 3: Substitute the known values: 27 = (1/2) * (3x) * ((1/3)x) Step 4: Simplify the right side: (1/2) * 3x * (1/3)x = (1/2) * (3 * 1/3) * x * x = (1/2) * 1 * x^2 = (1/2)x^2 Step 5: The equation becomes: 27 = (1/2)x^2 Step 6: Multiply both sides by 2: 54 = x^2 Step 7: Take the square root of both sides: x = sqrt(54) or x = 3*sqrt(6) or x is approximately 7.35 (since length must be positive). The answer is sqrt(54) or 3*sqrt(6) or approximately 7.35.

6. Tane is calculating the dimensions of a rectangular sports field for a community project. The length of the field is 12 meters more than three-quarters of its width. If the perimeter of the field is 192 meters, find the width of the field in meters. Answer: 48 Solution: Let w represent the width in meters. The length L is 12 more than three-quarters of the width: L = (3/4)w + 12. Step 2: The perimeter P of a rectangle is given by P = 2L + 2w.
   Full step-by-step solution

   Step 1: Let w represent the width in meters. The length L is 12 more than three-quarters of the width: L = (3/4)w + 12. Step 2: The perimeter P of a rectangle is given by P = 2L + 2w. We know P = 192, so: 2[(3/4)w + 12] + 2w = 192. Step 3: Distribute the 2: (3/2)w + 24 + 2w = 192. Step 4: Combine like terms: (3/2)w + (4/2)w = (7/2)w, so (7/2)w + 24 = 192. Step 5: Subtract 24 from both sides: (7/2)w = 168. Step 6: Multiply both sides by 2/7: w = 168 * (2/7) = 336/7 = 48. The width of the field is 48 meters.

7. 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.

8. A rectangle is drawn on a coordinate plane with vertices at (0,0), (x,0), (x, 2x/3), and (0, 2x/3). The perimeter of the rectangle is 50 units. Find the value of x. Answer: 15 Solution: Identify the dimensions from the coordinates. The length is along the x-axis from (0,0) to (x,0), so length = x. The width is along the y-axis from (0,0) to (0, 2x/3), so width = 2x/3.
   Full step-by-step solution

   Step 1: Identify the dimensions from the coordinates. The length is along the x-axis from (0,0) to (x,0), so length = x. The width is along the y-axis from (0,0) to (0, 2x/3), so width = 2x/3. Step 2: Write the perimeter formula for a rectangle. Perimeter = 2 × (length + width) Step 3: Substitute the known values. 50 = 2 × (x + 2x/3) Step 4: Simplify inside the parentheses. x + 2x/3 = (3x/3) + (2x/3) = 5x/3 Step 5: Continue solving. 50 = 2 × (5x/3) 50 = 10x/3 Step 6: Multiply both sides by 3. 150 = 10x Step 7: Divide both sides by 10. x = 15 The answer is 15.

9. 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.