To solve the given system of equations:
[tex]\[
\left\{
\begin{array}{l}
-3x + 4y = -19 \\
x - 4y = 17
\end{array}
\right.
\][/tex]
we will follow these steps:
### Step 1: Solve one of the equations for a single variable
Let's solve the second equation for [tex]\( x \)[/tex]:
[tex]\[ x - 4y = 17 \][/tex]
[tex]\[ x = 17 + 4y \][/tex]
### Step 2: Substitute this expression into the other equation
Now, substitute [tex]\( x = 17 + 4y \)[/tex] into the first equation:
[tex]\[ -3(17 + 4y) + 4y = -19 \][/tex]
### Step 3: Simplify and solve for [tex]\( y \)[/tex]
[tex]\[
-3(17 + 4y) + 4y = -19 \\
-51 - 12y + 4y = -19 \\
-51 - 8y = -19 \\
-8y = 32 \\
y = -4
\][/tex]
### Step 4: Substitute the value of [tex]\( y \)[/tex] back into the expression for [tex]\( x \)[/tex]
Now that we have [tex]\( y = -4 \)[/tex], substitute this value back into [tex]\( x = 17 + 4y \)[/tex]:
[tex]\[ x = 17 + 4(-4) \][/tex]
[tex]\[ x = 17 - 16 \][/tex]
[tex]\[ x = 1 \][/tex]
So, the solution to the system of equations is:
[tex]\[ x = 1, \quad y = -4 \][/tex]
### Step 5: Verify the solution
Substitute [tex]\( x = 1 \)[/tex] and [tex]\( y = -4 \)[/tex] back into both original equations to verify:
1. For the first equation:
[tex]\[ -3(1) + 4(-4) = -3 - 16 = -19 \][/tex]
2. For the second equation:
[tex]\[ 1 - 4(-4) = 1 + 16 = 17 \][/tex]
Both equations hold true, so the solution is correct.
One solution:
[tex]\[ (x, y) = (1, -4) \][/tex]
