To solve the system of linear equations given by:
[tex]\[
\begin{array}{l}
y = 4x + 3 \\
y = -x - 2
\end{array}
\][/tex]
we need to find the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that satisfy both equations simultaneously.
1. Step 1: Set the right-hand sides of the equations equal to each other because they both equal [tex]\(y\)[/tex].
[tex]\[
4x + 3 = -x - 2
\][/tex]
2. Step 2: Solve for [tex]\(x\)[/tex].
First, add [tex]\(x\)[/tex] to both sides to get:
[tex]\[
4x + x + 3 = -2
\][/tex]
which simplifies to:
[tex]\[
5x + 3 = -2
\][/tex]
Next, subtract 3 from both sides:
[tex]\[
5x = -2 - 3
\][/tex]
which simplifies to:
[tex]\[
5x = -5
\][/tex]
Finally, divide both sides by 5 to isolate [tex]\(x\)[/tex]:
[tex]\[
x = -1
\][/tex]
3. Step 3: Substitute the value of [tex]\(x\)[/tex] back into one of the original equations to solve for [tex]\(y\)[/tex]. We'll use the first equation [tex]\(y = 4x + 3\)[/tex]:
[tex]\[
y = 4(-1) + 3
\][/tex]
which simplifies to:
[tex]\[
y = -4 + 3
\][/tex]
[tex]\[
y = -1
\][/tex]
So, the solution to the system of equations is:
[tex]\[
(x, y) = (-1, -1)
\][/tex]
