LEAP - Louisiana Educational Assessment Program Solving Multi-Step Equations Questions and Answers 1 — Questions and Answers
Question 1: A student is solving the equation 5(x + 3) - 2x = 2(x + 9). The student's first step is to apply the distributive property, resulting in the equation 5x + 15 - 2x = 2x + 18. Which of the following would be the most logical next step to solve for x?
- Combine the like terms 5x and -2x on the left side of the equation. (Correct answer)
- Subtract 2x from both sides of the equation.
- Add 2x to both sides of the equation.
- Subtract 15 from both sides of the equation.
Correct answer: Combine the like terms 5x and -2x on the left side of the equation.
The most logical next step is to simplify the left side of the equation by combining like terms (5x and -2x). This simplifies the equation to 3x + 15 = 2x + 18, making it easier to isolate the variable in subsequent steps.
Question 2: Solve for y in the following equation: 8y - 4(y - 3) = 24.
- y = -9
- y = 9
- y = 3 (Correct answer)
- y = -3
Correct answer: y = 3
First, distribute the -4 to both terms inside the parentheses: 8y - 4y + 12 = 24. Next, combine like terms on the left side: 4y + 12 = 24. Then, subtract 12 from both sides: 4y = 12. Finally, divide both sides by 4 to find that y = 3.
Question 3: A rectangular garden has a perimeter of 78 feet. The length of the garden is 3 feet more than twice its width. If 'w' represents the width, which equation could be used to find the dimensions of the garden, and what is the correct width?
- 2(w) + 2(2w - 3) = 78; width = 14 feet
- w + (2w + 3) = 78; width = 25 feet
- 2(w) + (2w + 3) = 78; width = 18.75 feet
- 2(w) + 2(2w + 3) = 78; width = 12 feet (Correct answer)
Correct answer: 2(w) + 2(2w + 3) = 78; width = 12 feet
The formula for the perimeter of a rectangle is P = 2l + 2w. The length 'l' is given as 2w + 3. Substituting this into the perimeter formula gives 78 = 2(2w + 3) + 2w. Solving this equation: 78 = 4w + 6 + 2w, which simplifies to 78 = 6w + 6. Subtracting 6 gives 72 = 6w. Dividing by 6 gives w = 12 feet.
Question 4: What is the solution to the equation 1/2(8x - 10) = 3x + 7?
- x = -2
- x = 12 (Correct answer)
- x = 2
- x = 1.5
Correct answer: x = 12
First, apply the distributive property on the left side: 4x - 5 = 3x + 7. Next, subtract 3x from both sides to gather the variable terms: x - 5 = 7. Finally, add 5 to both sides to isolate x: x = 12.
Question 5: Which of the following equations has no solution?
- 5x + 4 = 5(x + 0.8)
- 7(x - 2) = 7x - 14
- 3x - 8 = 2x - 8
- 6(x + 1) = 6x - 2 (Correct answer)
Correct answer: 6(x + 1) = 6x - 2
To determine which equation has no solution, simplify each one. For 6(x + 1) = 6x - 2, distributing the 6 gives 6x + 6 = 6x - 2. If you subtract 6x from both sides, you get 6 = -2, which is a false statement. This indicates there is no value of x that will make the equation true, meaning it has no solution.
Question 6: The sum of three consecutive odd integers is 141. If 'n' is the smallest of the three integers, which equation represents this situation and what is the value of n?
- n + (n + 1) + (n + 2) = 141; n = 46
- n + (n + 2) + (n + 4) = 141; n = 45 (Correct answer)
- 3n + 3 = 141; n = 46
- 3n = 141; n = 47
Correct answer: n + (n + 2) + (n + 4) = 141; n = 45
Consecutive odd integers differ by 2. If the smallest integer is n, the next is n + 2, and the largest is n + 4. Their sum is n + (n + 2) + (n + 4) = 141. Combining like terms gives 3n + 6 = 141. Subtracting 6 from both sides results in 3n = 135. Dividing by 3 gives n = 45. The three integers are 45, 47, and 49.
LEAP - Louisiana Educational Assessment Program Solving Multi-Step Equations Questions and Answers 1
A student is solving the equation 5(x + 3) - 2x = 2(x + 9).
The student's first step is to apply the distributive property, resulting in the equation 5x + 15 - 2x = 2x + 18.
Which of the following would be the most logical next step to solve for x?