To solve the system of equations:
[tex]\[
\left\{
\begin{array}{l}
x + y = 4 \\
x = y - 1
\end{array}
\right.
\][/tex]
we can follow these steps:
### Step 1: Express one variable in terms of the other
From the second equation [tex]\( x = y - 1 \)[/tex], we've already expressed [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex].
### Step 2: Substitute this expression into the other equation
We now substitute [tex]\( x = y - 1 \)[/tex] into the first equation [tex]\( x + y = 4 \)[/tex]:
[tex]\[
(y - 1) + y = 4
\][/tex]
### Step 3: Simplify and solve for [tex]\( y \)[/tex]
Combine like terms in the equation from the previous step:
[tex]\[
2y - 1 = 4
\][/tex]
Add 1 to both sides:
[tex]\[
2y = 5
\][/tex]
Divide both sides by 2:
[tex]\[
y = \frac{5}{2}
\][/tex]
### Step 4: Solve for [tex]\( x \)[/tex]
Now, use the expression [tex]\( x = y - 1 \)[/tex] to find [tex]\( x \)[/tex]:
[tex]\[
x = \frac{5}{2} - 1 = \frac{5}{2} - \frac{2}{2} = \frac{3}{2}
\][/tex]
### Step 5: Verify the solution
Verify the solution by plugging [tex]\( x = \frac{3}{2} \)[/tex] and [tex]\( y = \frac{5}{2} \)[/tex] back into both original equations.
Substitute into [tex]\( x + y = 4 \)[/tex]:
[tex]\[
\frac{3}{2} + \frac{5}{2} = \frac{3+5}{2} = \frac{8}{2} = 4
\][/tex]
Substitute into [tex]\( x = y - 1 \)[/tex]:
[tex]\[
\frac{3}{2} = \frac{5}{2} - 1 = \frac{5}{2} - \frac{2}{2} = \frac{3}{2}
\][/tex]
Both original equations are satisfied.
Thus, the solution to the system of equations is:
[tex]\[
\left( \frac{3}{2}, \frac{5}{2}\right)
\][/tex]
When the choices are examined, the correct solution is indeed [tex]\(\left(\frac{3}{2}, \frac{5}{2}\right)\)[/tex].
