Question

Solve using the elimination method.

3x-5y=4
5x-3y=4

Answer 1

To solve the system of equations using the elimination method, we have the following equations:

1. \( 3x - 5y = 4 \) (Equation 1)
2. \( 5x - 3y = 4 \) (Equation 2)

First, we need to eliminate one of the variables. We can multiply each equation by suitable numbers to make the coefficients of \( y \) the same or to make them cancel out.

Let's eliminate \( y \). We can multiply Equation 1 by 3 and Equation 2 by 5:

- Multiply Equation 1 by 3:
\[
9x - 15y = 12 \quad \text{(Equation 3)}
\]

- Multiply Equation 2 by 5:
\[
25x - 15y = 20 \quad \text{(Equation 4)}
\]

Now we have:
1. \( 9x - 15y = 12 \) (Equation 3)
2. \( 25x - 15y = 20 \) (Equation 4)

Next, we can subtract Equation 3 from Equation 4 to eliminate \( y \):

\[
(25x - 15y) - (9x - 15y) = 20 - 12
\]

This simplifies to:

\[
(25x - 9x) = 8
\]
\[
16x = 8
\]
\[
x = \frac{8}{16} = \frac{1}{2}
\]

Now that we have the value of \( x \), we can substitute it back into one of the original equations to find \( y \). Let's use Equation 1:

\[
3\left(\frac{1}{2}\right) - 5y = 4
\]
\[
\frac{3}{2} - 5y = 4
\]

To isolate \( y \), first subtract \( \frac{3}{2} \) from both sides:

\[
-5y = 4 - \frac{3}{2}
\]

To perform the subtraction, convert \( 4 \) to a fraction with a denominator of \( 2 \):

\[
4 = \frac{8}{2}
\]
So,
\[
-5y = \frac{8}{2} - \frac{3}{2} = \frac{5}{2}
\]

Now divide by \(-5\):

\[
y = \frac{5/2}{-5} = -\frac{1}{2}
\]

Thus, the solution to the system of equations is:

\[
x = \frac{1}{2}, \quad y = -\frac{1}{2}
\]

The final answer is:
\[
\left( x, y \right) = \left( \frac{1}{2}, -\frac{1}{2} \right)
\]