To solve the system of linear equations using the linear combination method (or elimination method), we proceed as follows:
### Given System of Equations:
[tex]\[
\begin{cases}
2d - e = 7 \\
d + e = 5
\end{cases}
\][/tex]
### Step-by-Step Solution:
1. Write down the equations:
[tex]\[
\text{Equation (1): } 2d - e = 7
\][/tex]
[tex]\[
\text{Equation (2): } d + e = 5
\][/tex]
2. Choose one variable to eliminate:
We'll eliminate [tex]\(e\)[/tex].
3. Add the two equations together to eliminate [tex]\(e\)[/tex]:
Let's add Equation (1) and Equation (2):
[tex]\[
(2d - e) + (d + e) = 7 + 5
\][/tex]
4. Simplify the resulting equation:
[tex]\[
2d - e + d + e = 12
\][/tex]
[tex]\[
3d = 12
\][/tex]
5. Solve for [tex]\(d\)[/tex]:
[tex]\[
d = \frac{12}{3} = 4
\][/tex]
6. Substitute the value of [tex]\(d\)[/tex] back into one of the original equations:
We use Equation (2) for this:
[tex]\[
4 + e = 5
\][/tex]
7. Solve for [tex]\(e\)[/tex]:
[tex]\[
e = 5 - 4 = 1
\][/tex]
So, we find:
[tex]\[
d = 4, \quad e = 1
\][/tex]
### Conclusion:
The solution to the system of equations is [tex]\((d, e) = (4,1)\)[/tex]. This can be written as:
[tex]\[
\boxed{(4,1)}
\][/tex]
