Question

Which linear equation has the same solution set as `\frac{9x}{4}+3-2x=-2\left(3+x\right)?`

Answer 1

We start by simplifying the equation given:

\[
\frac{9x}{4} + 3 - 2x = -2(3 + x)
\]

First, we can simplify the right-hand side:

\[
-2(3 + x) = -6 - 2x
\]

Now we can rewrite the equation:

\[
\frac{9x}{4} + 3 - 2x = -6 - 2x
\]

Next, we can add \(2x\) to both sides to eliminate \(-2x\) from both sides:

\[
\frac{9x}{4} + 3 = -6
\]

Now, we subsequently need to isolate \(\frac{9x}{4}\). We can subtract 3 from both sides:

\[
\frac{9x}{4} = -6 - 3
\]
\[
\frac{9x}{4} = -9
\]

Next, we multiply both sides by 4 to eliminate the fraction:

\[
9x = -9 \times 4
\]

Calculating the right-hand side:

\[
9x = -36
\]

Finally, we divide both sides by 9:

\[
x = \frac{-36}{9} = -4
\]

Now that we have the solution \(x = -4\), we want to construct a linear equation that has the same solution set. The simplest form of a linear equation with the same solution would be:

\[
x + 4 = 0
\]

Alternatively, we can express this in a standard form:

\[
x = -4
\]

This can be re-written as:

\[
x + 4 = 0
\]

Thus, the linear equation that has the same solution set as the original is:

\[
x + 4 = 0
\]