To solve the system of linear equations:
1. [tex]\(3x - y = 11\)[/tex]
2. [tex]\(6x - 3y = 18\)[/tex]
we can approach it step-by-step as follows:
1. First equation:
[tex]\[
3x - y = 11 \quad \text{(Equation 1)}
\][/tex]
2. Second equation:
[tex]\[
6x - 3y = 18 \quad \text{(Equation 2)}
\][/tex]
3. We can simplify Equation 2 by dividing every term by 3:
[tex]\[
\frac{6x}{3} - \frac{3y}{3} = \frac{18}{3}
\][/tex]
Simplifying, we get:
[tex]\[
2x - y = 6 \quad \text{(Simplified Equation 2)}
\][/tex]
4. Now, we have a new system of equations:
- [tex]\(3x - y = 11\)[/tex]
- [tex]\(2x - y = 6\)[/tex]
5. To eliminate [tex]\(y\)[/tex], we subtract the second equation from the first:
[tex]\[
(3x - y) - (2x - y) = 11 - 6
\][/tex]
Simplifying this, we get:
[tex]\[
x = 5
\][/tex]
6. With [tex]\(x = 5\)[/tex], we substitute this value back into one of our simplified equations to find [tex]\(y\)[/tex]. Let's use [tex]\(2x - y = 6\)[/tex]:
[tex]\[
2(5) - y = 6
\][/tex]
Simplifying:
[tex]\[
10 - y = 6
\][/tex]
Solving for [tex]\(y\)[/tex]:
[tex]\[
y = 10 - 6
\][/tex]
[tex]\[
y = 4
\][/tex]
Thus, the solution to the system of equations is:
[tex]\[
\boxed{x = 5, y = 4}
\][/tex]
