To solve the given system of linear equations:
[tex]\[
\begin{cases}
y = 3x - 4 \\
y = -\frac{1}{2}x + 3
\end{cases}
\][/tex]
we can use the method of setting the [tex]\( y \)[/tex]-values equal to each other since both equations are equal to [tex]\( y \)[/tex].
1. Set the equations equal to each other:
[tex]\[
3x - 4 = -\frac{1}{2}x + 3
\][/tex]
2. Combine like terms:
First, let's get all the [tex]\( x \)[/tex]-terms on one side of the equation. We can start by adding [tex]\(\frac{1}{2}x\)[/tex] to both sides:
[tex]\[
3x + \frac{1}{2}x - 4 = 3
\][/tex]
Combine the [tex]\( x \)[/tex]-terms on the left side. Note that [tex]\( 3x = \frac{6}{2}x \)[/tex], so:
[tex]\[
\frac{6}{2}x + \frac{1}{2}x - 4 = 3
\][/tex]
Combine the fractions:
[tex]\[
\frac{7}{2}x - 4 = 3
\][/tex]
3. Isolate the [tex]\( x \)[/tex]-term:
Add 4 to both sides of the equation:
[tex]\[
\frac{7}{2}x = 3 + 4
\][/tex]
Simplify the right side:
[tex]\[
\frac{7}{2}x = 7
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
To isolate [tex]\( x \)[/tex], multiply both sides by the reciprocal of [tex]\(\frac{7}{2}\)[/tex], which is [tex]\(\frac{2}{7}\)[/tex]:
[tex]\[
x = 7 \cdot \frac{2}{7}
\][/tex]
Simplify:
[tex]\[
x = 2
\][/tex]
5. Substitute [tex]\( x \)[/tex] back into one of the original equations to solve for [tex]\( y \)[/tex]:
Substitute [tex]\( x = 2 \)[/tex] into the first equation [tex]\( y = 3x - 4 \)[/tex]:
[tex]\[
y = 3(2) - 4
\][/tex]
[tex]\[
y = 6 - 4
\][/tex]
[tex]\[
y = 2
\][/tex]
Therefore, the solution to the system of equations is:
[tex]\[
(x, y) = (2, 2)
\][/tex]
So, the point of intersection of the two lines, and thus the solution to the system, is [tex]\( (2.0, 2.0) \)[/tex].
