To solve the system of equations using the elimination method, we start with the given equations:
- \( 3x - 4y = -4 \)
- \( x + 3y = -10 \)
We will eliminate \( x \) or \( y \) by manipulating these equations. In this case, let's eliminate \( x \).
First, we can multiply the second equation by 3 so that the coefficients of \( x \) in both equations become the same:
\[ 3(x + 3y) = 3(-10) \ 3x + 9y = -30 \tag{3} \]
Now we have the modified system:
- \( 3x - 4y = -4 \)
- \( 3x + 9y = -30 \)
Next, we can eliminate \( x \) by subtracting equation 1 from equation 3:
\[ (3x + 9y) - (3x - 4y) = -30 - (-4) \ 3x + 9y - 3x + 4y = -30 + 4 \ 13y = -26 \]
Now, solve for \( y \):
\[ y = \frac{-26}{13} = -2 \]
Now that we have \( y \), we can substitute this value back into one of the original equations to find \( x \). We'll use the second equation \( x + 3y = -10 \):
\[ x + 3(-2) = -10 \ x - 6 = -10 \ x = -10 + 6 \ x = -4 \]
So the solution to the system of equations is:
\[ (x, y) = (-4, -2) \]
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