Solving Multi-Step Equations Flashcards

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?

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.

Solve for y in the following equation: 8y - 4(y - 3) = 24.

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.

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?

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.

What is the solution to the equation 1/2(8x - 10) = 3x + 7?

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.

Which of the following equations has no solution?

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.

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?

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.