To find the solution to the system of equations:
[tex]\[
\begin{cases}
y = 2x + 3 \\
y = -x + 9
\end{cases}
\][/tex]
we need to determine the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that satisfy both equations simultaneously. Let's solve this step-by-step.
Step 1: Set the two equations equal to each other since they both equal [tex]\(y\)[/tex].
[tex]\[
2x + 3 = -x + 9
\][/tex]
Step 2: Combine like terms to isolate [tex]\(x\)[/tex]. First, add [tex]\(x\)[/tex] to both sides of the equation:
[tex]\[
2x + x + 3 = 9
\][/tex]
This simplifies to:
[tex]\[
3x + 3 = 9
\][/tex]
Step 3: Subtract 3 from both sides of the equation to isolate the term with [tex]\(x\)[/tex]:
[tex]\[
3x = 6
\][/tex]
Step 4: Divide both sides by 3 to solve for [tex]\(x\)[/tex]:
[tex]\[
x = 2
\][/tex]
Step 5: Substitute the value of [tex]\(x = 2\)[/tex] back into one of the original equations to find the corresponding value of [tex]\(y\)[/tex]. We'll use the first equation [tex]\(y = 2x + 3\)[/tex]:
[tex]\[
y = 2(2) + 3
\][/tex]
This simplifies to:
[tex]\[
y = 4 + 3 \implies y = 7
\][/tex]
Therefore, the solution to the system of equations is:
[tex]\[
(x, y) = (2, 7)
\][/tex]
So, the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that satisfy both equations are [tex]\(x = 2\)[/tex] and [tex]\(y = 7\)[/tex].
