To solve the system of equations:
- \( 2x + 7y = -17 \) (Equation 1)
- \( -6x - 10y = -4 \) (Equation 2)
We'll use the substitution or elimination method. First, let's simplify Equation 2.
Multiply Equation 2 by -1 to get:
\[ 6x + 10y = 4 \tag{Equation 3} \]
Now, we will work with Equations 1 and 3. We'll use the elimination method. To eliminate \(x\), we can manipulate these two equations. First, we need to adjust the coefficients of \(x\) in both equations. We can multiply Equation 1 by 3:
\[ 3(2x + 7y) = 3(-17) \] \[ 6x + 21y = -51 \tag{Equation 4} \]
Now we will subtract Equation 3 from Equation 4:
\[ (6x + 21y) - (6x + 10y) = -51 - 4 \] \[ 6x + 21y - 6x - 10y = -55 \] \[ 11y = -55 \] \[ y = -5 \]
Now that we have the value of \(y\), we can substitute it back into Equation 1 to find \(x\):
\[ 2x + 7(-5) = -17 \] \[ 2x - 35 = -17 \] \[ 2x = -17 + 35 \] \[ 2x = 18 \] \[ x = 9 \]
So, the solution is:
\[ x = 9, \quad y = -5 \]
Thus, the solution to the system of equations is \((x, y) = (9, -5)\).
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