Sure, let's solve the system of equations step by step.
The given system of equations is:
[tex]\[
\left\{
\begin{array}{l}
14x - 11y = -29 \\
13y - 8x = 30
\end{array}
\right.
\][/tex]
### Step 1: Solve for one variable
Let's start by solving one of the equations for one variable in terms of the other.
From the first equation:
[tex]\[ 14x - 11y = -29 \][/tex]
We can solve for [tex]\(x\)[/tex]:
[tex]\[ 14x = 11y - 29 \][/tex]
[tex]\[ x = \frac{11y - 29}{14} \][/tex]
### Step 2: Substitute this expression into the second equation
Substitute [tex]\( x = \frac{11y - 29}{14} \)[/tex] into the second equation:
[tex]\[ 13y - 8 \left( \frac{11y - 29}{14} \right) = 30 \][/tex]
### Step 3: Simplify the equation
First, we distribute [tex]\(-8\)[/tex]:
[tex]\[ 13y - \frac{88y - 232}{14} = 30 \][/tex]
Clear the fraction by multiplying everything by 14:
[tex]\[ 14 \cdot 13y - 88y + 232 = 14 \cdot 30 \][/tex]
[tex]\[ 182y - 88y + 232 = 420 \][/tex]
Combine like terms:
[tex]\[ 94y + 232 = 420 \][/tex]
### Step 4: Solve for y
Subtract 232 from both sides:
[tex]\[ 94y = 188 \][/tex]
Divide by 94:
[tex]\[ y = \frac{188}{94} \][/tex]
[tex]\[ y = 2 \][/tex]
### Step 5: Substitute y back into the expression for x
Now that we have [tex]\( y = 2 \)[/tex], we substitute it back into the expression for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{11(2) - 29}{14} \][/tex]
[tex]\[ x = \frac{22 - 29}{14} \][/tex]
[tex]\[ x = \frac{-7}{14} \][/tex]
[tex]\[ x = -\frac{1}{2} \][/tex]
### Solution
The solution to the system of equations is:
[tex]\[ x = -\frac{1}{2} \][/tex]
[tex]\[ y = 2 \][/tex]
So, the solution pair [tex]\((x, y)\)[/tex] is [tex]\(\left( -\frac{1}{2}, 2 \right)\)[/tex].
