Question

Study the equations.

Information
Equation I: 5 x plus 3 y equals 19



Equation II: 2 x minus 4 y equals negative 8

Question
In solving the system of equations, which operations will allow the elimination of the variable x in the next step?

Answer options with 5 options
A.
Multiply Equation I by 1-half and multiply Equation II by 1-fifth.

B.
Multiply Equation I by 5 and multiply Equation II by 2.

C.
Multiply Equation I by 2 and multiply Equation II by negative 5.

D.
Multiply Equation I by 4 and multiply Equation II by 3.

E.
Multiply both equations by 10.

Answer 1

To eliminate the variable \( x \) in the given system of equations, we need to manipulate the coefficients of \( x \) in such a way that they are opposites in the two equations.

Let’s look at the equations:

1. **Equation I**: \( 5x + 3y = 19 \)
2. **Equation II**: \( 2x - 4y = -8 \)

The coefficient of \( x \) in Equation I is \( 5 \) and in Equation II it is \( 2 \). To eliminate \( x \), we can make the coefficients equal and opposite.

One way to do this is to find a common multiple of \( 5 \) and \( 2 \). The least common multiple of \( 5 \) and \( 2 \) is \( 10 \). Therefore, we will try to make the coefficient of \( x \) in both equations equal to \( 10 \):

- If we multiply Equation I by \( 2 \), we get:
\[
2(5x + 3y) = 2(19) \implies 10x + 6y = 38
\]

- If we multiply Equation II by \( -5 \), we get:
\[
-5(2x - 4y) = -5(-8) \implies -10x + 20y = 40
\]

Now, we have:
1. \( 10x + 6y = 38 \)
2. \( -10x + 20y = 40 \)

These two equations can now be combined to eliminate \( x \).

Therefore, the correct answer is option:

**C. Multiply Equation I by 2 and multiply Equation II by negative 5.**