To solve the system of equations:
[tex]\[
\begin{cases}
x + 2y + 5z = 9 \\
x = y + 8 \\
z = x - 3
\end{cases}
\][/tex]
we need to find the values of [tex]\( x \)[/tex], [tex]\( y \)[/tex], and [tex]\( z \)[/tex] that satisfy all three equations simultaneously. Let's proceed step by step:
### Step 1: Express [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]
From the second equation:
[tex]\[
x = y + 8
\][/tex]
### Step 2: Express [tex]\( z \)[/tex] in terms of [tex]\( x \)[/tex]
From the third equation:
[tex]\[
z = x - 3
\][/tex]
### Step 3: Substitute [tex]\( x \)[/tex] and [tex]\( z \)[/tex] into the first equation
Substitute [tex]\( x = y + 8 \)[/tex] and [tex]\( z = x - 3 \)[/tex] into the first equation:
[tex]\[
x + 2y + 5z = 9
\][/tex]
Replace [tex]\( x \)[/tex] with [tex]\( y + 8 \)[/tex]:
[tex]\[
(y + 8) + 2y + 5z = 9
\][/tex]
Replace [tex]\( z \)[/tex] with [tex]\( x - 3 \)[/tex] which is [tex]\( (y + 8) - 3 \)[/tex]:
[tex]\[
(y + 8) + 2y + 5((y + 8) - 3) = 9
\][/tex]
Simplify [tex]\( z \)[/tex]:
[tex]\[
z = y + 5
\][/tex]
Now substitute [tex]\( z = y + 5 \)[/tex] back into the equation:
[tex]\[
(y + 8) + 2y + 5(y + 5) = 9
\][/tex]
### Step 4: Simplify and solve for [tex]\( y \)[/tex]
Combine like terms:
[tex]\[
y + 8 + 2y + 5y + 25 = 9
\][/tex]
Combine all the [tex]\( y \)[/tex] terms:
[tex]\[
8y + 33 = 9
\][/tex]
Subtract 33 from both sides:
[tex]\[
8y = -24
\][/tex]
Divide both sides by 8:
[tex]\[
y = -3
\][/tex]
### Step 5: Solve for [tex]\( x \)[/tex] and [tex]\( z \)[/tex]
Using [tex]\( y = -3 \)[/tex] in the equation [tex]\( x = y + 8 \)[/tex]:
[tex]\[
x = -3 + 8
\][/tex]
[tex]\[
x = 5
\][/tex]
Now using [tex]\( x = 5 \)[/tex] in the equation [tex]\( z = x - 3 \)[/tex]:
[tex]\[
z = 5 - 3
\][/tex]
[tex]\[
z = 2
\][/tex]
### Solution
The values of the variables that satisfy all the equations are:
[tex]\[
x = 5, \ y = -3, \ z = 2
\][/tex]
