How many solution(s) does the system of equations have?

Question

−5x+y=−18

10x−2y=36
(1 point)
Responses

two
two

one
one

infinite
infinite

none

Answer 1

To determine how many solutions the system of equations has, we can examine the equations:

1. \(-5x + y = -18\)
2. \(10x - 2y = 36\)

First, let's rearrange the first equation to express \(y\) in terms of \(x\):

From the first equation: \[ y = 5x - 18 \]

Next, simplify the second equation: \[ 10x - 2y = 36 \]

We can also rearrange this second equation to express \(y\) in terms of \(x\). Start by isolating \(2y\): \[ 2y = 10x - 36 \] Now divide everything by 2: \[ y = 5x - 18 \]

Now we have:

1. \(y = 5x - 18\)
2. \(y = 5x - 18\)

Both equations are identical. Since they represent the same line, every point on that line is a solution to the system of equations.

Thus, the system of equations has infinite solutions.

The correct response is infinite.