Question

equations with variables on both sides
the work below shows student’s steps to solve equations with variables on both sides. in each of the problems, an error is made in one line of work. under the “find it” column, circle the line of work where there is an error made. under the “fix it” column, rework the equation to show the correct steps and solution.

1. -5x - 16 = 9x + x - 1

-5x - 16 = 8x - 1
-16 = 13x - 1
-15 = 13x
x = -\frac{15}{13}
22 + 53x + x = -20 + 60x
22 + 54x = -20 + 60x
22 = -20 + 6x
2 = 6x
x = \frac{1}{3}

Explanation:

Step1: Identify the error in the first equation

The error is in the second - line of the first equation. When combining like - terms on the right - hand side of $-5x - 16=9x + x - 1$, $9x+x$ should be $10x$, not $8x$.

Step2: Correct the first equation

Starting with $-5x - 16=9x + x - 1$, which simplifies to $-5x - 16 = 10x-1$. Add $5x$ to both sides: $-16=10x + 5x-1$, so $-16 = 15x-1$. Then add 1 to both sides: $-16 + 1=15x$, which gives $-15 = 15x$. Divide both sides by 15: $x=-1$.

Step3: Identify the error in the second equation

The error is in the fourth - line of the second equation. When going from $22=-20 + 6x$ to the next step, 20 should be added to both sides, not subtracted.

Step4: Correct the second equation

Starting with $22 + 53x+x=-20 + 60x$, which simplifies to $22+54x=-20 + 60x$. Subtract $54x$ from both sides: $22=-20+60x - 54x$, so $22=-20 + 6x$. Add 20 to both sides: $22 + 20=6x$, which gives $42 = 6x$. Divide both sides by 6: $x = 7$.

Answer:

For the first equation:

• Error line: $-5x - 16 = 8x - 1$
• Correct solution: $x=-1$

For the second equation:

• Error line: $2 = 6x$
• Correct solution: $x = 7$