To solve the system of linear equations:
[tex]\[
\begin{cases}
2x - 5y = -21 \quad \text{(Equation 1)} \\
3x + 5y = 6 \quad \text{(Equation 2)}
\end{cases}
\][/tex]
we need to find the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that satisfy both equations simultaneously.
### Step 1: Write the equations
First, let's restate the given equations for clarity:
[tex]\[ 2x - 5y = -21 \][/tex]
[tex]\[ 3x + 5y = 6 \][/tex]
### Step 2: Add the equations
To eliminate one of the variables, we can add the two equations. Notice that the coefficients of [tex]\(y\)[/tex] are [tex]\(-5\)[/tex] and [tex]\(5\)[/tex], which will cancel each other out when added:
[tex]\[
(2x - 5y) + (3x + 5y) = -21 + 6
\][/tex]
Simplifying the left-hand side and right-hand side:
[tex]\[
(2x + 3x) + (-5y + 5y) = -21 + 6
\][/tex]
[tex]\[
5x = -15
\][/tex]
### Step 3: Solve for [tex]\(x\)[/tex]
Now, solve for [tex]\(x\)[/tex] by dividing both sides by 5:
[tex]\[
x = \frac{-15}{5}
\][/tex]
[tex]\[
x = -3
\][/tex]
### Step 4: Substitute [tex]\(x = -3\)[/tex] into one of the original equations
Next, we substitute [tex]\(x = -3\)[/tex] into one of the original equations to find [tex]\(y\)[/tex]. We’ll use Equation 1:
[tex]\[
2(-3) - 5y = -21
\][/tex]
Simplify:
[tex]\[
-6 - 5y = -21
\][/tex]
### Step 5: Solve for [tex]\(y\)[/tex]
Add 6 to both sides to isolate the term with [tex]\(y\)[/tex]:
[tex]\[
-5y = -21 + 6
\][/tex]
[tex]\[
-5y = -15
\][/tex]
Divide both sides by -5:
[tex]\[
y = \frac{-15}{-5}
\][/tex]
[tex]\[
y = 3
\][/tex]
### Step 6: Verify the solution
To confirm our solution, we substitute [tex]\(x = -3\)[/tex] and [tex]\(y = 3\)[/tex] into both original equations to ensure they hold true.
For Equation 1:
[tex]\[
2(-3) - 5(3) = -21
\][/tex]
[tex]\[
-6 - 15 = -21
\][/tex]
[tex]\[
-21 = -21 \quad \text{(True)}
\][/tex]
For Equation 2:
[tex]\[
3(-3) + 5(3) = 6
\][/tex]
[tex]\[
-9 + 15 = 6
\][/tex]
[tex]\[
6 = 6 \quad \text{(True)}
\][/tex]
Both equations are satisfied, indicating that our solution [tex]\( (x, y) = (-3, 3) \)[/tex] is correct.
Thus, the answer is:
[tex]\[ \boxed{(-3, 3)} \][/tex]
