Question

Solve the system using the elimination method.

5x-2y=1
3x-4y=-19

Answer 1

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

1) \( 5x - 2y = 1 \)
2) \( 3x - 4y = -19 \)

First, we can manipulate these equations to eliminate one of the variables. Let's eliminate \( y \) by making the coefficients of \( y \) the same in both equations.

To do this, we can multiply the first equation by \( 2 \) to match the coefficient of \( y \) in the second equation:

\[
2(5x - 2y) = 2(1)
\]

This yields:

\[
10x - 4y = 2 \quad \text{(Equation 3)}
\]

Now we have:

1) \( 10x - 4y = 2 \) (Equation 3)
2) \( 3x - 4y = -19 \) (Equation 2)

Next, we can subtract Equation 2 from Equation 3 to eliminate \( y \):

\[
(10x - 4y) - (3x - 4y) = 2 - (-19)
\]

Simplifying this gives:

\[
10x - 4y - 3x + 4y = 2 + 19
\]

This simplifies to:

\[
7x = 21
\]

Now, divide both sides by \( 7 \):

\[
x = 3
\]

Now that we have the value for \( x \), we can substitute it back into one of the original equations to solve for \( y \). We'll use the first equation:

\[
5x - 2y = 1
\]

Substituting \( x = 3 \):

\[
5(3) - 2y = 1
\]

This simplifies to:

\[
15 - 2y = 1
\]

Now, isolate \( y \):

\[
-2y = 1 - 15
\]
\[
-2y = -14
\]

Dividing by \( -2 \):

\[
y = 7
\]

So the solution to the system of equations is:

\[
x = 3, \quad y = 7
\]

Thus, the solution is \( (3, 7) \).