Answer :
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.
[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.
