To solve the given system of equations:
[tex]\[
\begin{cases}
y = 2x + 3 \\
y = -x + 9
\end{cases}
\][/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 process to solve this:
1. Set the equations equal to each other: Since both equations equal [tex]\(y\)[/tex], we can set the right-hand sides of the equations equal to each other:
[tex]\[
2x + 3 = -x + 9
\][/tex]
2. Isolate [tex]\(x\)[/tex]: To solve for [tex]\(x\)[/tex], we need to get all the [tex]\(x\)[/tex] terms on one side and the constant terms on the other side. Start by adding [tex]\(x\)[/tex] to both sides to move the [tex]\(x\)[/tex] terms to one side:
[tex]\[
2x + x + 3 = 9
\][/tex]
Simplify:
[tex]\[
3x + 3 = 9
\][/tex]
3. Solve for [tex]\(x\)[/tex]: Subtract 3 from both sides to isolate the [tex]\(3x\)[/tex] term:
[tex]\[
3x = 6
\][/tex]
Divide both sides by 3:
[tex]\[
x = 2
\][/tex]
4. Substitute [tex]\(x\)[/tex] back into one of the original equations: We can use either equation to find the corresponding value of [tex]\(y\)[/tex]. Let's use the first equation [tex]\(y = 2x + 3\)[/tex]:
Substitute [tex]\(x = 2\)[/tex]:
[tex]\[
y = 2(2) + 3
\][/tex]
Simplify:
[tex]\[
y = 4 + 3
\][/tex]
Therefore:
[tex]\[
y = 7
\][/tex]
5. Verify the solution: To ensure correctness, substitute [tex]\(x = 2\)[/tex] and [tex]\(y = 7\)[/tex] into the second original equation [tex]\(y = -x + 9\)[/tex]:
[tex]\[
7 = -(2) + 9
\][/tex]
Simplify:
[tex]\[
7 = 7
\][/tex]
The solution satisfies both equations.
Thus, the solution to the system of equations is:
[tex]\[
\boxed{x = 2, y = 7}
\][/tex]
