To solve the system of linear equations:
[tex]\[
\begin{cases}
7x + 4y = 13 \\
5x - 2y = 19
\end{cases}
\][/tex]
we can use the method of substitution or elimination. We'll go through the steps using the elimination method for clarity.
### Step 1: Align the Equations
Write the equations clearly:
1. [tex]\(7x + 4y = 13\)[/tex]
2. [tex]\(5x - 2y = 19\)[/tex]
### Step 2: Eliminate One Variable
To eliminate [tex]\(y\)[/tex], we can multiply the second equation by 2 so that the [tex]\(y\)[/tex]-terms in both equations have the same coefficients (with opposite signs for cancellation).
Multiplying the second equation by 2:
[tex]\[2(5x - 2y) = 2 \cdot 19\][/tex]
which simplifies to:
[tex]\[10x - 4y = 38\][/tex]
### Step 3: Add the Equations
Now we have the system:
1. [tex]\(7x + 4y = 13\)[/tex]
2. [tex]\(10x - 4y = 38\)[/tex]
Add these two equations:
[tex]\[
(7x + 4y) + (10x - 4y) = 13 + 38
\][/tex]
which simplifies to:
[tex]\[
17x = 51
\][/tex]
### Step 4: Solve for [tex]\(x\)[/tex]
Divide both sides of the equation by 17:
[tex]\[
x = \frac{51}{17} = 3
\][/tex]
### Step 5: Substitute [tex]\(x\)[/tex] Back Into One of the Original Equations
Substitute [tex]\(x = 3\)[/tex] into the first original equation:
[tex]\[
7(3) + 4y = 13
\][/tex]
which simplifies to:
[tex]\[
21 + 4y = 13
\][/tex]
### Step 6: Solve for [tex]\(y\)[/tex]
Subtract 21 from both sides:
[tex]\[
4y = 13 - 21 \rightarrow 4y = -8
\][/tex]
Divide by 4:
[tex]\[
y = \frac{-8}{4} = -2
\][/tex]
### Final Solution
The solution to the system of equations is:
[tex]\[
x = 3 \quad \text{and} \quad y = -2
\][/tex]
So, the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that satisfy both equations are [tex]\(x = 3\)[/tex] and [tex]\(y = -2\)[/tex].
