To solve the system of equations:
[tex]\[
\left\{\begin{array}{l}
3x + 5y = 0 \\
3x - 2y = 21
\end{array}\right.
\][/tex]
we will use the method of elimination or substitution to find the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
### Step 1: Arrange the Equations
First, we write down the two equations clearly:
1. [tex]\( 3x + 5y = 0 \)[/tex]
2. [tex]\( 3x - 2y = 21 \)[/tex]
### Step 2: Eliminate One Variable
We will eliminate [tex]\( x \)[/tex] by subtracting one equation from the other. To do this, we can first subtract equation 1 from equation 2:
[tex]\[
(3x - 2y) - (3x + 5y) = 21 - 0
\][/tex]
Simplifying the left-hand side:
[tex]\[
3x - 2y - 3x - 5y = 21
\][/tex]
[tex]\[
-7y = 21
\][/tex]
### Step 3: Solve for [tex]\( y \)[/tex]
We solve this equation for [tex]\( y \)[/tex]:
[tex]\[
y = \frac{21}{-7}
\][/tex]
[tex]\[
y = -3
\][/tex]
### Step 4: Substitute [tex]\( y \)[/tex] Back to Solve for [tex]\( x \)[/tex]
Next, we substitute [tex]\( y = -3 \)[/tex] back into one of the original equations. Let's use the first equation for this purpose:
[tex]\[
3x + 5(-3) = 0
\][/tex]
[tex]\[
3x - 15 = 0
\][/tex]
[tex]\[
3x = 15
\][/tex]
[tex]\[
x = 5
\][/tex]
### Step 5: Write the Solution
Therefore, the solution to the system of equations is:
[tex]\[
x = 5, \quad y = -3
\][/tex]
Thus, [tex]\((x, y) = (5.0, -3.0)\)[/tex].
