To find the solution to the system of equations given by:
- \( x - 5y = 3 \) (Equation 1)
- \( 3x - 2y = -4 \) (Equation 2)
Let's first solve each equation for \( y \) in terms of \( x \) to facilitate graphing and understanding.
Step 1: Solve Equation 1 for \( y \)
Starting with Equation 1:
\[ x - 5y = 3 \]
Rearranging it to isolate \( y \):
\[ -5y = 3 - x \]
Now, divide both sides by -5:
\[ y = \frac{1}{5}x - \frac{3}{5} \]
Step 2: Solve Equation 2 for \( y \)
Now for Equation 2:
\[ 3x - 2y = -4 \]
Rearranging to solve for \( y \):
\[ -2y = -4 - 3x \]
Divide both sides by -2:
\[ y = \frac{3}{2}x + 2 \]
Step 3: Find the Intersection Point
We can now set the two equations for \( y \) equal to each other to find the intersection point:
\[ \frac{1}{5}x - \frac{3}{5} = \frac{3}{2}x + 2 \]
To eliminate the fractions, multiply everything by 10:
\[ 2x - 6 = 15x + 20 \]
Now simplify the equation:
\[ 2x - 15x = 20 + 6 \]
\[ -13x = 26 \]
Now, solve for \( x \):
\[ x = -2 \]
Step 4: Substitute \( x \) back to find \( y \)
Now substitute \( x = -2 \) back into either original equation. We can use Equation 1:
\[ -2 - 5y = 3 \]
Rearranging gives:
\[ -5y = 3 + 2 \]
\[ -5y = 5 \]
Now solve for \( y \):
\[ y = -1 \]
Conclusion
The solution to the system of equations \( x - 5y = 3 \) and \( 3x - 2y = -4 \) is:
\[ \boxed{(-2, -1)} \]
Graphing
If we were to graph these two equations, we would plot the lines represented by each equation. The intersection point \((-2, -1)\) would be the point where both lines cross, representing the solution to this system of equations.
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