Let's solve the given system of linear equations step-by-step.
The system of equations is:
1. [tex]\( x + 2y = 6 \)[/tex]
2. [tex]\( x - y = 3 \)[/tex]
To find the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex], we can use the method of substitution or elimination. Here, we'll use the substitution method.
### Step 1: Solve the second equation for [tex]\( x \)[/tex]
From the second equation, [tex]\( x - y = 3 \)[/tex], we can solve for [tex]\( x \)[/tex]:
[tex]\[ x = y + 3 \][/tex]
### Step 2: Substitute [tex]\( x \)[/tex] in the first equation
Now, substitute [tex]\( x = y + 3 \)[/tex] into the first equation [tex]\( x + 2y = 6 \)[/tex]:
[tex]\[ (y + 3) + 2y = 6 \][/tex]
### Step 3: Simplify and solve for [tex]\( y \)[/tex]
Combine like terms:
[tex]\[ y + 3 + 2y = 6 \][/tex]
[tex]\[ 3y + 3 = 6 \][/tex]
Subtract 3 from both sides:
[tex]\[ 3y = 3 \][/tex]
Divide both sides by 3:
[tex]\[ y = 1 \][/tex]
### Step 4: Solve for [tex]\( x \)[/tex]
Now that we have [tex]\( y = 1 \)[/tex], we can use the expression [tex]\( x = y + 3 \)[/tex] to find [tex]\( x \)[/tex]:
[tex]\[ x = 1 + 3 \][/tex]
[tex]\[ x = 4 \][/tex]
### Solution
Thus, the solution to the system of equations is:
[tex]\[ x = 4 \][/tex]
[tex]\[ y = 1 \][/tex]
The values [tex]\( x = 4 \)[/tex] and [tex]\( y = 1 \)[/tex] satisfy both equations in the system.
