Question

Solving systems by elimination 7x-4y=27 -14x+3y=-29

Answer 1

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

1. \( 7x - 4y = 27 \) (Equation 1)
2. \( -14x + 3y = -29 \) (Equation 2)

First, we want to align the coefficients of one of the variables so that we can eliminate it. We can multiply Equation 1 by 2 to align the coefficients of \(x\):

\[
2(7x - 4y) = 2(27)
\]

This gives us:

\[
14x - 8y = 54 \quad \text{(Equation 3)}
\]

Now, we have:

1. \( 14x - 8y = 54 \) (Equation 3)
2. \( -14x + 3y = -29 \) (Equation 2)

Next, we will add Equation 3 and Equation 2 to eliminate \(x\):

\[
(14x - 8y) + (-14x + 3y) = 54 + (-29)
\]

This simplifies to:

\[
-5y = 25
\]

Now, we will solve for \(y\):

\[
y = \frac{25}{-5} = -5
\]

Now that we have \(y = -5\), we will substitute this value back into one of the original equations to find \(x\). We'll use Equation 1:

\[
7x - 4(-5) = 27
\]

This simplifies to:

\[
7x + 20 = 27
\]

Subtract 20 from both sides:

\[
7x = 27 - 20
\]
\[
7x = 7
\]

Now, divide both sides by 7:

\[
x = 1
\]

The solution to the system of equations is:

\[
x = 1, \quad y = -5
\]

Thus, the solution is \( (1, -5) \).