To find the solution to the system of equations:
[tex]\[
\begin{array}{l}
y=-\frac{1}{2} x+9 \\
y=x+7
\end{array}
\][/tex]
we need to determine the point where these two lines intersect.
1. Set the equations equal to each other because [tex]\( y \)[/tex] is the same in both equations at the point of intersection:
[tex]\[
-\frac{1}{2} x + 9 = x + 7
\][/tex]
2. Solve for [tex]\( x \)[/tex]:
- Move [tex]\( x \)[/tex] terms to one side of the equation:
[tex]\[
9 - 7 = x + \frac{1}{2} x
\][/tex]
- Simplify the equation:
[tex]\[
2 = \frac{3}{2} x
\][/tex]
- Multiply both sides by [tex]\(\frac{2}{3}\)[/tex] to isolate [tex]\( x \)[/tex]:
[tex]\[
x = \frac{2 \times 2}{3} = \frac{4}{3}
\][/tex]
3. Solve for [tex]\( y \)[/tex] using one of the original equations, substituting [tex]\( x = \frac{4}{3} \)[/tex]:
[tex]\[
y = x + 7 \\
y = \frac{4}{3} + 7
\][/tex]
- Convert 7 to a fraction with a denominator of 3:
[tex]\[
y = \frac{4}{3} + \frac{21}{3} = \frac{25}{3}
\][/tex]
Thus, the point of intersection is:
[tex]\[
\left( \frac{4}{3}, \frac{25}{3} \right)
\][/tex]
Expressing this in decimal form for clarity:
[tex]\[
\left( 1.33333333333333, 8.33333333333333 \right)
\][/tex]
Given the options:
- Lines [tex]\(y=-\frac{1}{2} x+9\)[/tex] and [tex]\(y=x+7\)[/tex] intersect the [tex]\(x\)[/tex]-axis.
- Lines [tex]\(y=-\frac{1}{2} x+9\)[/tex] and [tex]\(y=x+7\)[/tex] intersect the [tex]\(y\)[/tex]-axis.
- Line [tex]\(y=-\frac{1}{2} x+9\)[/tex] intersects the origin.
- Line [tex]\(y=-\frac{1}{2} x+9\)[/tex] intersects line [tex]\(y=x+7\)[/tex].
The best description of this solution is:
Line [tex]\( y = -\frac{1}{2} x + 9 \)[/tex] intersects line [tex]\( y = x + 7 \)[/tex].
