To solve the given system of equations:
[tex]\[
\begin{array}{l}
y = -\frac{1}{2} x + 9 \\
y = x + 7
\end{array}
\][/tex]
we need to find the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that satisfy both equations simultaneously.
Here is the step-by-step solution:
1. Set the equations equal to each other: Since both expressions are set equal to [tex]\(y\)[/tex], we can set the right-hand sides of the equations equal to each other:
[tex]\[
-\frac{1}{2} x + 9 = x + 7
\][/tex]
2. Solve for [tex]\(x\)[/tex]:
- First, isolate the [tex]\(x\)[/tex] terms on one side of the equation. Adding [tex]\(\frac{1}{2}x\)[/tex] to both sides yields:
[tex]\[
9 = \frac{3}{2} x + 7
\][/tex]
- Subtract 7 from both sides to further isolate the [tex]\(x\)[/tex] term:
[tex]\[
2 = \frac{3}{2} x
\][/tex]
- Multiply both sides of the equation by [tex]\(\frac{2}{3}\)[/tex] to solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{2}{3} \cdot \frac{3}{2} = \frac{4}{3} \approx 1.33333333333333
\][/tex]
3. Solve for [tex]\(y\)[/tex]:
- Substitute the value of [tex]\(x\)[/tex] back into either of the original equations. Using [tex]\(y = x + 7\)[/tex]:
[tex]\[
y = \frac{4}{3} + 7
\][/tex]
- Convert 7 to a fraction with a common denominator:
[tex]\[
7 = \frac{21}{3}
\][/tex]
- Add the fractions:
[tex]\[
y = \frac{4}{3} + \frac{21}{3} = \frac{25}{3} \approx 8.33333333333333
\][/tex]
4. Conclusion:
- The solution to the system of equations is [tex]\( x \approx 1.33333333333333 \)[/tex] and [tex]\( y \approx 8.33333333333333 \)[/tex].
Therefore, the description that best describes the solution to the system of equations is:
[tex]\[\{x: 1.33333333333333, y: 8.33333333333333\}\][/tex]
