Answer :

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]