To solve the system of linear equations
[tex]\[
\begin{cases}
4x - 3y = 7 \\
5x - 2y = 0
\end{cases}
\][/tex]
we will solve it step-by-step.
### Step 1: Write the equations
1. [tex]\( 4x - 3y = 7 \)[/tex]
2. [tex]\( 5x - 2y = 0 \)[/tex]
### Step 2: Solve one of the equations for one variable
From the second equation:
[tex]\[ 5x - 2y = 0 \][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[ 5x = 2y \][/tex]
[tex]\[ x = \frac{2}{5}y \][/tex]
### Step 3: Substitute [tex]\( x \)[/tex] into the first equation
Now, substitute [tex]\( x = \frac{2}{5}y \)[/tex] into the first equation:
[tex]\[ 4 \left(\frac{2}{5}y\right) - 3y = 7 \][/tex]
[tex]\[ \frac{8}{5}y - 3y = 7 \][/tex]
### Step 4: Simplify and solve for [tex]\( y \)[/tex]
[tex]\[ \frac{8}{5}y - \frac{15}{5}y = 7 \][/tex]
[tex]\[ \frac{8 - 15}{5}y = 7 \][/tex]
[tex]\[ \frac{-7}{5}y = 7 \][/tex]
Multiply both sides by [tex]\( -\frac{5}{7} \)[/tex]:
[tex]\[ y = 7 \times -\frac{5}{7} \][/tex]
[tex]\[ y = -5 \][/tex]
### Step 5: Substitute [tex]\( y \)[/tex] back into the equation for [tex]\( x \)[/tex]
Using [tex]\( y = -5 \)[/tex] in [tex]\( x = \frac{2}{5}y \)[/tex]:
[tex]\[ x = \frac{2}{5}(-5) \][/tex]
[tex]\[ x = -2 \][/tex]
### Conclusion
The solution to the system of equations is [tex]\( x = -2 \)[/tex] and [tex]\( y = -5 \)[/tex].
