To solve the system of equations given by:
[tex]\[
\begin{array}{l}
7x - 4y = -8 \\
y = \frac{3}{4}x - 3
\end{array}
\][/tex]
we follow these steps:
1. Substitute the expression for [tex]\(y\)[/tex] from the second equation into the first equation:
The second equation gives [tex]\( y = \frac{3}{4}x - 3 \)[/tex].
Substitute [tex]\( y = \frac{3}{4}x - 3 \)[/tex] into the first equation:
[tex]\[
7x - 4\left( \frac{3}{4}x - 3 \right) = -8
\][/tex]
2. Simplify the equation:
Distribute [tex]\( -4 \)[/tex] across the terms inside the parentheses:
[tex]\[
7x - 4 \cdot \frac{3}{4}x + 4 \cdot 3 = -8
\][/tex]
This simplifies to:
[tex]\[
7x - 3x + 12 = -8
\][/tex]
Combine like terms:
[tex]\[
4x + 12 = -8
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
Isolate [tex]\( x \)[/tex] by first subtracting 12 from both sides:
[tex]\[
4x = -8 - 12
\][/tex]
[tex]\[
4x = -20
\][/tex]
Divide both sides by 4:
[tex]\[
x = -5
\][/tex]
4. Substitute [tex]\( x \)[/tex] back into the second original equation to find [tex]\( y \)[/tex]:
Use the second equation [tex]\( y = \frac{3}{4}x - 3 \)[/tex]:
[tex]\[
y = \frac{3}{4}(-5) - 3
\][/tex]
Multiply:
[tex]\[
y = -\frac{15}{4} - 3
\][/tex]
Convert [tex]\(-3\)[/tex] to quarters:
[tex]\[
y = -\frac{15}{4} - \frac{12}{4}
\][/tex]
Add the fractions:
[tex]\[
y = -\frac{27}{4}
\][/tex]
Convert [tex]\(-\frac{27}{4}\)[/tex] to a decimal:
[tex]\[
y = -6.75
\][/tex]
Therefore, the solution to the system of equations is approximately:
[tex]\[
(x, y) = (-5.0, -6.75)
\][/tex]
This is the point where the two equations intersect on the graph.
