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Shannon graphed the system of equations.

[tex]\[
\begin{array}{l}
7x - 4y = -8 \\
y = \frac{3}{4}x - 3
\end{array}
\][/tex]

What is the closest approximate solution to the system of equations?

Sagot :

GinnyAnswer
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.
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