A variable needs to be eliminated to solve the system of equations below. Choose the correct first step. $\newline$ $\begin{array}{l} 5 x+2 y=23 \\ 5 x+3 y=32 \end{array}$ $\newline$ Add to eliminate $\mathbf{x}$ . $\newline$ Subtract to eliminate $\mathbf{x}$ . $\newline$ Add to eliminate $\mathbf{y}$ . $\newline$ Subtract to eliminate $\mathbf{y}$ .

Full solution

Q. A variable needs to be eliminated to solve the system of equations below. Choose the correct first step.

\newline

\begin{array}{l} 5 x+2 y=23 \\ 5 x+3 y=32 \end{array}

\newline

Add to eliminate

\mathbf{x}

\newline

Subtract to eliminate

\mathbf{x}

\newline

Add to eliminate

\mathbf{y}

\newline

Subtract to eliminate

\mathbf{y}

Analyze Coefficients: Analyze the coefficients of $x$ and $y$ in both equations. $\newline$ We have the system of equations: $\newline$ $5x + 2y = 23$ $\newline$ $5x + 3y = 32$ $\newline$ To eliminate a variable, we need to make the coefficients of either $x$ or $y$ opposite in both equations so that when we add or subtract the equations, one variable cancels out.
Compare Coefficients of $x$ : Compare the coefficients of $x$ in both equations. $\newline$ The coefficients of $x$ in both equations are the same ( $5$ and $5$ ). This means we can eliminate $x$ by subtracting one equation from the other.
Compare Coefficients of $y$ : Compare the coefficients of $y$ in both equations. $\newline$ The coefficients of $y$ are $2$ and $3$ , which are not the same and not opposites, so we cannot directly eliminate $y$ by adding or subtracting the equations without first multiplying one or both equations by a number to make the coefficients of $y$ opposites.
Decide on Operation: Decide on the operation to eliminate $x$ . Since the coefficients of $x$ are already the same, we can subtract the second equation from the first to eliminate $x$ . $(5x + 2y) - (5x + 3y) = 23 - 32$ This will eliminate $x$ because $5x - 5x = 0$ .