Linear with Rationals

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Answer Key & Explanations

Linear with Rationals · Grade 9 · Worksheet 2

1. Liam is designing a rectangular garden where the length is 3 meters more than twice the width. The area of the garden must be 65 square meters to accommodate his vegetable plants. 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 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 m², so: \[ (2w + 3) \times w = 65 \] --- **Step 3: Expand and rearrange** \[ 2w^2 + 3w = 65 \] \[ 2w^2 + 3w - 65 = 0 \] --- **Step 4: Solve the quadratic equation** We can use the quadratic formula: \( w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 2 \), \( b = 3 \), \( c = -65 \). First, discriminant \( D = b^2 - 4ac \): \[ D = 3^2 - 4(2)(-65) = 9 + 520 = 529 \] \[ \sqrt{D} = \sqrt{529} = 23 \] --- **Step 5: Apply quadratic formula** \[ w = \frac{-3 \pm 23}{2 \times 2} = \frac{-3 \pm 23}{4} \] Two possible solutions: \[ w = \frac{-3 + 23}{4} = \frac{20}{4} = 5 \] \[ w = \frac{-3 - 23}{4} = \frac{-26}{4} = -6.5 \] --- **Step 6: Interpret the solutions** Width cannot be negative, so \( w = -6.5 \) is not valid. Thus, \( w = 5 \) meters. --- **Step 7: Check** Width = 5 m Length = \( 2 \times 5 + 3 = 13 \) m Area = \( 13 \times 5 = 65 \) m² ✅ --- **Final answer:** The width is 5 meters.

2. 2(x + 3) - 5 = 3(x - 1) + 4 = ? Answer: x = 2 Solution: Linear equations with rational coefficients require careful application of the distributive property and inverse operations. When solving equations like 3(y + 2) = 2(y - 1), you would distribute first, then use addition or subtraction to isolate the variable term, and finally divide to find the…
   Full step-by-step solution

   Linear equations with rational coefficients require careful application of the distributive property and inverse operations. When solving equations like 3(y + 2) = 2(y - 1), you would distribute first, then use addition or subtraction to isolate the variable term, and finally divide to find the solution.

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

   When solving linear equations with rational coefficients, first apply the distributive property to remove parentheses. Then combine constant terms and variable terms separately on each side. Finally, use inverse operations to isolate the variable, maintaining balance by performing the same operation on both sides of the equation.

4. Noah is calculating the total resistance in an electrical circuit with two resistors connected in parallel. The total resistance R (in ohms) satisfies the equation 2/3R + 1/4 = 7/12. Solve for R to find the total resistance in ohms. Answer: 0.5 Solution: Start with the equation 2/3R + 1/4 = 7/12. Subtract 1/4 from both sides: 2/3R = 7/12 - 1/4. Convert 1/4 to twelfths: 1/4 = 3/12.
   Full step-by-step solution

   Step 1: Start with the equation 2/3R + 1/4 = 7/12. Step 2: Subtract 1/4 from both sides: 2/3R = 7/12 - 1/4. Step 3: Convert 1/4 to twelfths: 1/4 = 3/12. Step 4: Subtract: 7/12 - 3/12 = 4/12 = 1/3. Step 5: Now the equation is 2/3R = 1/3. Step 6: Multiply both sides by the reciprocal of 2/3, which is 3/2: R = (1/3) * (3/2) = 1/2. Step 7: R = 1/2 ohm = 0.5 ohm. The answer is 0.5.

5. Liam is designing a rectangular garden with a length that is 3 meters more than twice its width. He needs to install a decorative border along the perimeter, which requires 36 meters of material. What are the dimensions of Liam's garden? Answer: width = 5 meters, length = 13 meters Solution: Let the width of the garden be \( w \) meters. The length is 3 meters more than twice the width, so: length \( l = 2w + 3 \). \( P = 2 \times (\text{length} + \text{width}) \).
   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 perimeter equation** The perimeter \( P \) of a rectangle is: \( P = 2 \times (\text{length} + \text{width}) \). We are told the perimeter is 36 meters: \( 2 \times (l + w) = 36 \). --- **Step 3: Substitute the expression for length** Substitute \( l = 2w + 3 \) into the perimeter equation: \( 2 \times ( (2w + 3) + w ) = 36 \). --- **Step 4: Simplify inside the parentheses** \( (2w + 3) + w = 3w + 3 \). So: \( 2 \times (3w + 3) = 36 \). --- **Step 5: Solve for \( w \)** Divide both sides by 2: \( 3w + 3 = 18 \). Subtract 3 from both sides: \( 3w = 15 \). Divide by 3: \( w = 5 \). --- **Step 6: Find length** \( l = 2w + 3 = 2(5) + 3 = 10 + 3 = 13 \). --- **Step 7: Final answer** Width = 5 meters, Length = 13 meters. --- **Check:** Perimeter = \( 2 \times (13 + 5) = 2 \times 18 = 36 \), which matches the problem.

6. 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

7. A right triangle is drawn on a coordinate plane with vertices at (0,0), (x,0), and (0,2x). The area of the triangle is 36 square units. Find the value of x. Answer: 6 Solution: Identify the base and height of the triangle. The vertices are (0,0), (x,0), and (0,2x). The segment from (0,0) to (x,0) is horizontal, so its length is x.
   Full step-by-step solution

   Step 1: Identify the base and height of the triangle. The vertices are (0,0), (x,0), and (0,2x). The segment from (0,0) to (x,0) is horizontal, so its length is x. The segment from (0,0) to (0,2x) is vertical, so its length is 2x. Step 2: Recognize that in a right triangle with legs along the axes, these two segments are the base and height. Let base = x and height = 2x. Step 3: Write the formula for the area of a triangle. Area = (1/2) * base * height Step 4: Substitute the known values into the formula. Area = (1/2) * x * (2x) Step 5: Simplify the expression. (1/2) * x * 2x = (1/2) * 2 * x * x = 1 * x^2 = x^2 Step 6: Set the area equal to 36. x^2 = 36 Step 7: Solve for x. x = sqrt(36) x = 6 (since length must be positive) Final answer: x = 6

8. (3x - 7)/4 - (2x + 5)/3 = 11/12 Answer: x = 14 Solution: Identify the denominators: 4, 3, and 12. The least common denominator is 12.
   Full step-by-step solution

   Step 1: Identify the denominators: 4, 3, and 12. The least common denominator is 12. Step 2: Multiply both sides of the equation by 12: 12 * [(3x - 7)/4 - (2x + 5)/3] = 12 * (11/12) Step 3: Distribute the 12 to each term on the left: 12 * (3x - 7)/4 - 12 * (2x + 5)/3 = 11 Step 4: Simplify each fraction: 3(3x - 7) - 4(2x + 5) = 11 Step 5: Distribute the coefficients: 9x - 21 - 8x - 20 = 11 Step 6: Combine like terms: (9x - 8x) + (-21 - 20) = 11 x - 41 = 11 Step 7: Add 41 to both sides: x = 52 The answer is x = 52.

9. Noah is calculating the total cost of a road trip. The car uses fuel at a rate of 0.06 gallons per mile. Gas costs $3.50 per gallon. If the total fuel cost for the trip is $126, how many miles is the trip? Answer: 600 Solution: Let x be the number of miles in the trip. Fuel used = 0.06x gallons. Total fuel cost = (0.06x)(3.50) = 126.
   Full step-by-step solution

   Step 1: Let x be the number of miles in the trip. Step 2: Fuel used = 0.06x gallons. Step 3: Total fuel cost = (0.06x)(3.50) = 126. Step 4: Simplify: 0.21x = 126. Step 5: Divide both sides by 0.21: x = 126 / 0.21 = 600. The answer is 600.