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$\begin{aligned} 4x-4y &= -2 \\ -9x-4y &= -3 \end{aligned}$

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Find equations of tangent lines using limits

Full solution

Q.

\begin{aligned} 4x-4y &= -2 \\ -9x-4y &= -3 \end{aligned}

Write Equations: First, let's write down the system of equations we need to solve: $\newline$ \begin{aligned} $\newline$ $4$ x - $4$ y &= $-2$ (\newline\) $-9$ x - $4$ y &= $-3$ $\newline$ \end{aligned}
Elimination Method: To solve the system, we can use the method of elimination. We will try to eliminate one of the variables by adding or subtracting the equations. First, we need to make the coefficients of one of the variables the same in both equations. In this case, the coefficients of $y$ are already the same, so we can proceed to add the two equations to eliminate $y$ .
Add Equations: Now, let's add the two equations: $\newline$ \begin{aligned} $\newline$ ( $4$ x - $4$ y) + ( $-9$ x - $4$ y) &= $-2$ + ( $-3$ ) (\newline\)( $4$ x - $9$ x) + ( $-4$ y - $4$ y) &= $-2$ - $3$ (\newline\) $-5$ x - $8$ y &= $-5$ $\newline$ \end{aligned} $\newline$ It seems we have made a mistake in adding the equations. The $y$ terms should cancel out, but they didn't because we added them instead of canceling them. Let's correct this error.

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