To solve the system of linear equations:
[tex]\[
\begin{array}{l}
2x + 7y = -1 \\
4x - 3y = -19
\end{array}
\][/tex]
we will use the method of elimination step by step.
### Step 1: Align the equations
[tex]\[
\begin{array}{l}
2x + 7y = -1 \quad \text{(Equation 1)} \\
4x - 3y = -19 \quad \text{(Equation 2)}
\end{array}
\][/tex]
### Step 2: Eliminate one variable
First, we want to eliminate \( x \). We can do this by making the coefficients of \( x \) the same in both equations. To do this, we can multiply Equation 1 by 2.
[tex]\[
4x + 14y = -2 \quad \text{(Equation 3, derived from Equation 1)}
\][/tex]
Now we have:
[tex]\[
\begin{array}{l}
4x + 14y = -2 \quad \text{(Equation 3)} \\
4x - 3y = -19 \quad \text{(Equation 2)}
\end{array}
\][/tex]
### Step 3: Subtract the equations
Next, subtract Equation 2 from Equation 3 to eliminate \( x \):
[tex]\[
(4x + 14y) - (4x - 3y) = -2 - (-19)
\][/tex]
Which simplifies to:
[tex]\[
4x + 14y - 4x + 3y = 17
\][/tex]
[tex]\[
17y = 17
\][/tex]
### Step 4: Solve for \( y \)
Divide both sides by 17:
[tex]\[
y = 1
\][/tex]
### Step 5: Substitute \( y \) back into one of the original equations
Let’s substitute \( y = 1 \) into Equation 1:
[tex]\[
2x + 7(1) = -1
\][/tex]
[tex]\[
2x + 7 = -1
\][/tex]
[tex]\[
2x = -1 - 7
\][/tex]
[tex]\[
2x = -8
\][/tex]
[tex]\[
x = -4
\][/tex]
### Step 6: Write the solution as an ordered pair
Thus, the solution to the system of equations is:
[tex]\[
(x, y) = (-4, 1)
\][/tex]
