Question

Solving systems by elimination 2x-4y=-20 5x-3y=-29

Answer 1

To solve the system of equations using the elimination method, we start with the following two equations:

1. \( 2x - 4y = -20 \) (Equation 1)
2. \( 5x - 3y = -29 \) (Equation 2)

### Step 1: Align the coefficients
We want to eliminate one of the variables. Let's eliminate \( y \). To do so, we can manipulate the equations so that the coefficients of \( y \) are the same.

First, we can multiply Equation 1 by \( \frac{3}{4} \) so that the coefficient of \( y \) becomes the same in both equations. This means multiplying throughout by \( 3 \):

\[
3(2x - 4y) = 3(-20)
\]

This gives us:

\[
6x - 12y = -60 \quad \text{(Equation 1 modified)}
\]

Now, we leave Equation 2 as is:

\[
5x - 3y = -29 \quad \text{(Equation 2)}
\]

### Step 2: Scale the second equation

To eliminate \( y \), we will now scale Equation 2 so that the coefficients of \( y \) match. We can multiply Equation 2 by \( 4 \):

\[
4(5x - 3y) = 4(-29)
\]

This gives us:

\[
20x - 12y = -116 \quad \text{(Equation 2 modified)}
\]

### Step 3: Set the new equations up for elimination

Now we have the system:

1. \( 6x - 12y = -60 \)
2. \( 20x - 12y = -116 \)

### Step 4: Eliminate \( y \)

Now subtract Equation 1 from Equation 2:

\[
(20x - 12y) - (6x - 12y) = -116 - (-60)
\]

This simplifies to:

\[
14x = -56
\]

### Step 5: Solve for \( x \)

Now we can solve for \( x \):

\[
x = \frac{-56}{14} = -4
\]

### Step 6: Substitute back to find \( y \)

Now that we have \( x \), we can substitute \( x = -4 \) back into one of the original equations to solve for \( y \). We'll use Equation 1:

\[
2(-4) - 4y = -20
\]

This becomes:

\[
-8 - 4y = -20
\]

Adding \( 8 \) to both sides:

\[
-4y = -20 + 8
\]
\[
-4y = -12
\]

Now divide by \( -4 \):

\[
y = \frac{-12}{-4} = 3
\]

### Final solution

The solution to the system of equations is:

\[
x = -4, \quad y = 3
\]

Thus, the ordered pair is:

\[
\boxed{(-4, 3)}
\]