Create Equations

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Answer Key & Explanations

Create Equations · Grade 9 · Worksheet 3

1. Emma is designing a rectangular banner for a school event. The banner has a length that is 5 meters more than twice its width. Write an equation that relates the length (L) and width (W) of the banner. Answer: L = 2W + 5 Solution: When creating equations from word problems, first identify the variables involved and how they relate to each other.
   Full step-by-step solution

   When creating equations from word problems, first identify the variables involved and how they relate to each other. Look for phrases that indicate mathematical operations, such as 'more than' for addition or 'twice' for multiplication.

2. Olivia is designing a rectangular garden with a perimeter of 50 meters. If the length is 5 meters more than twice the width, write an equation in two variables that represents this situation. Answer: P = 2l + 2w, l = 2w + 5 Solution: When creating equations from geometric contexts, identify the standard formulas involved and then incorporate the given relationships between variables.
   Full step-by-step solution

   When creating equations from geometric contexts, identify the standard formulas involved and then incorporate the given relationships between variables. For example, if a rectangle's area is fixed and one dimension is a multiple of the other, you would use the area formula and substitute the relationship.

3. Noah is designing a rectangular mural for a school project. The length of the mural is 6 feet longer than its width. The total area of the mural is 216 square feet. Write an equation in two variables to represent the relationship between the width (w) and the length (l) of the mural, and then write a second equation that relates the area to the width and length. Use these equations to find the dimensions of the mural. Answer: Width = 12 feet, Length = 18 feet Solution: Let w represent the width and l represent the length. From the problem, the length is 6 feet longer than the width, so l = w + 6. Step 2: The area of a rectangle is given by A = l * w.
   Full step-by-step solution

   Step 1: Let w represent the width and l represent the length. From the problem, the length is 6 feet longer than the width, so l = w + 6. Step 2: The area of a rectangle is given by A = l * w. We know the area is 216, so l * w = 216. Step 3: Substitute l = w + 6 into the area equation: (w + 6) * w = 216. Step 4: Expand: w^2 + 6w = 216. Step 5: Rearrange to form a quadratic equation: w^2 + 6w - 216 = 0. Step 6: Factor the quadratic: (w + 18)(w - 12) = 0. Step 7: Solve for w: w = -18 or w = 12. Since width cannot be negative, w = 12 feet. Step 8: Find the length: l = w + 6 = 12 + 6 = 18 feet. The dimensions are width 12 feet and length 18 feet.

4. Matiu is designing a rectangular garden with a path around it. The garden itself has length L meters and width W meters. The path surrounding the garden has a uniform width of 2 meters on all sides. Write an equation for the total area A of the garden plus the path, in terms of L and W. Answer: A = (L + 4)(W + 4) Solution: The garden is a rectangle with length L and width W. A path of uniform width 2 meters surrounds the garden on all sides.
   Full step-by-step solution

   Step 1: The garden is a rectangle with length L and width W. Step 2: A path of uniform width 2 meters surrounds the garden on all sides. This means the path adds 2 meters to the left, 2 meters to the right, 2 meters to the top, and 2 meters to the bottom. Step 3: Therefore, the total length of the garden plus path is L + 2 + 2 = L + 4. Step 4: The total width of the garden plus path is W + 2 + 2 = W + 4. Step 5: The total area A (garden + path) is the area of the outer rectangle: total length times total width. Step 6: So the equation is A = (L + 4)(W + 4). The answer is A = (L + 4)(W + 4).

5. Liam is designing a rectangular garden with a perimeter of 40 meters. He wants the length to be 4 meters more than twice the width. Write a system of equations to represent this situation and solve for the dimensions of the garden. Answer: width = 6 meters, length = 14 meters Solution: When solving geometry problems with unknown dimensions, we often use variables to represent the measurements we're trying to find.
   Full step-by-step solution

   When solving geometry problems with unknown dimensions, we often use variables to represent the measurements we're trying to find. The perimeter formula for rectangles provides one equation, while the relationship between length and width gives us another. By substituting one equation into the other, we can solve for the unknown values systematically.

6. (2x² - 5x - 3) ÷ (x - 3) = ? Answer: 2x + 1 Solution: Set up polynomial long division with (2x² - 5x - 3) ÷ (x - 3) Divide the first term: 2x² ÷ x = 2x Multiply: 2x × (x - 3) = 2x² - 6x Subtract: (2x² - 5x - 3) - (2x² - 6x) = x - 3 Divide the next term: x ÷ x = 1 Multiply: 1 × (x - 3) = x - 3 Subtract: (x - 3) - (x - 3) = 0 The quotient is 2x + 1…
   Full step-by-step solution

   Step 1: Set up polynomial long division with (2x² - 5x - 3) ÷ (x - 3) Step 2: Divide the first term: 2x² ÷ x = 2x Step 3: Multiply: 2x × (x - 3) = 2x² - 6x Step 4: Subtract: (2x² - 5x - 3) - (2x² - 6x) = x - 3 Step 5: Divide the next term: x ÷ x = 1 Step 6: Multiply: 1 × (x - 3) = x - 3 Step 7: Subtract: (x - 3) - (x - 3) = 0 Step 8: The quotient is 2x + 1 with remainder 0 The answer is 2x + 1.

7. Sophia is designing a rectangular garden with a perimeter of 36 meters. The length is 6 meters more than the width. Write an equation in two variables for the perimeter. Answer: 2(l + w) = 36 Solution: When creating equations from geometric contexts, identify the key measurements and their relationships.
   Full step-by-step solution

   When creating equations from geometric contexts, identify the key measurements and their relationships. The perimeter formula for rectangles uses both length and width, and additional information about how these dimensions relate to each other helps create a complete mathematical model.

8. 2x² - 5x + 3 = 0 Answer: x = 1, x = 1.5 Solution: 2x² - 5x + 3 = 0 a x² + b x + c = 0 Here: a = 2, b = -5, c = 3 We look for two numbers that multiply to a * c = 2 * 3 = 6 and add to b = -5. The numbers -2 and -3 multiply to 6 but add to -5? No: (-2) + (-3) = -5, yes.
   Full step-by-step solution

   We are solving the quadratic equation: 2x² - 5x + 3 = 0 --- **Step 1: Identify coefficients** The equation is in the form: a x² + b x + c = 0 Here: a = 2, b = -5, c = 3 --- **Step 2: Check if factoring is possible** We look for two numbers that multiply to a * c = 2 * 3 = 6 and add to b = -5. The numbers -2 and -3 multiply to 6 but add to -5? No: (-2) + (-3) = -5, yes. So we split the middle term -5x into -2x - 3x. --- **Step 3: Factor by grouping** 2x² - 5x + 3 = 2x² - 2x - 3x + 3 Group terms: (2x² - 2x) + (-3x + 3) Factor each group: 2x(x - 1) - 3(x - 1) Now factor out (x - 1): (x - 1)(2x - 3) = 0 --- **Step 4: Solve for x** Set each factor equal to 0: x - 1 = 0 → x = 1 2x - 3 = 0 → 2x = 3 → x = 3/2 = 1.5 --- **Final Answer:** x = 1, x = 1.5