Question

How many solution(s) does the system of equations have?
−5x+y=−18

10x−2y=36
(1 point)
Responses

two
two

one
one

none
none

infinite

Answer 1

To determine how many solutions the system of equations has, we can analyze the two equations given:

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

We can rearrange both equations into slope-intercept form (y = mx + b) or analyze them directly.

First, let's simplify the second equation:

From \( 10x - 2y = 36 \), we can isolate \( y \):

\[
-2y = -10x + 36
\]

\[
y = 5x - 18
\]

Now we can write the first equation in slope-intercept form:

From \( -5x + y = -18 \), rearranging gives:

\[
y = 5x - 18
\]

Now we have:

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

Both equations are equivalent (they produce the same line), meaning they represent the same line on a graph.

Therefore, the system has **infinite solutions** because every point on the line is a solution to the system of equations.

The answer is **infinite**.