To solve the given system of equations:
[tex]\[
\begin{cases}
3x + 4y = 17 \\
-4x - 3y = -18
\end{cases}
\][/tex]
we can use the method of elimination. The goal is to eliminate one of the variables by adding or subtracting the equations.
First, let's multiply the first equation by 3 and the second equation by 4 to make the coefficients of [tex]\( y \)[/tex] the same (but opposite in sign).
Multiplying the first equation by 3:
[tex]\[ 3(3x + 4y) = 3 \cdot 17 \][/tex]
This simplifies to:
[tex]\[ 9x + 12y = 51 \][/tex]
Multiplying the second equation by 4:
[tex]\[ 4(-4x - 3y) = 4 \cdot (-18) \][/tex]
This simplifies to:
[tex]\[ -16x - 12y = -72 \][/tex]
Now, we add the two resulting equations to eliminate [tex]\( y \)[/tex]:
[tex]\[
\begin{aligned}
(9x + 12y) + (-16x - 12y) &= 51 + (-72) \\
(9x - 16x) + (12y - 12y) &= -21 \\
-7x &= -21
\end{aligned}
\][/tex]
Solving for [tex]\( x \)[/tex], we divide both sides by -7:
[tex]\[
x = \frac{-21}{-7} = 3
\][/tex]
So, [tex]\( x = 3 \)[/tex].
Next, we substitute [tex]\( x = 3 \)[/tex] back into one of the original equations to solve for [tex]\( y \)[/tex]. Using the first equation:
[tex]\[ 3(3) + 4y = 17 \][/tex]
[tex]\[ 9 + 4y = 17 \][/tex]
[tex]\[ 4y = 17 - 9 \][/tex]
[tex]\[ 4y = 8 \][/tex]
[tex]\[ y = \frac{8}{4} = 2 \][/tex]
So, [tex]\( y = 2 \)[/tex].
Therefore, the solution to the system of equations is [tex]\((x, y) = (3, 2)\)[/tex].
Expressed as an ordered pair:
[tex]\[
(3, 2)
\][/tex]
