Question

Which equations are equivalent? I. \(7 x-9=7\) II. \(-9=7-5 x\) III. \(3(2 x-3)=7-x\) A. I and III B. II and III C. All D. None

Short answer

The equivalent equations are I and III, so the answer is A. I and III.

Step-by-step solution

Arrange Equation I in Standard Form

The standard form of a linear equation is \( ax = b \). Arrange the first equation \( 7x - 9 = 7 \) into this form. This can be done by adding 9 to both sides, resulting in \( 7x = 16 \).

Arrange Equation II in Standard Form

Arrange the second equation \( -9 = 7 - 5x \) into the standard form. This can be done by adding 5x to both sides and then subtracting 7 from both sides, resulting in \( 5x = 16 \).

Arrange Equation III in Standard Form

Arrange the third equation \( 3(2x - 3) = 7 -x \) into the standard form. Start by expanding the brackets to get \( 6x - 9 = 7 - x \). Then, add x to both sides and add 9 to both sides to get \( 7x = 16 \).

Comparison of Equations

Compare the equations from Steps 1, 2 and 3 to see which are equivalent. From the comparison, it can be seen that Equation I and Equation III are equivalent since they both result in \( 7x = 16 \). Equation II is set to \( 5x = 16 \), so it's not equivalent to Equation I and III.