Question

Assume a system of equations contains the lines y=23x−3
and y=−13x+3
. What is the solution for this system?(1 point)
Responses

There are infinite solutions.
There are infinite solutions.

(6,1)
left parenthesis 6 comma 1 right parenthesis

There is no solution.
There is no solution.

(0,3)

Answer 1

To find the solution to the system of equations given by the lines \( y = 23x - 3 \) and \( y = -13x + 3 \), we need to find the point where the two lines intersect.

We can set the equations equal to each other:

\[
23x - 3 = -13x + 3
\]

Now, let's solve for \( x \):

1. Add \( 13x \) to both sides:

\[
23x + 13x - 3 = 3
\]

\[
36x - 3 = 3
\]

2. Add 3 to both sides:

\[
36x = 6
\]

3. Divide by 36:

\[
x = \frac{6}{36} = \frac{1}{6}
\]

Now that we have \( x = \frac{1}{6} \), we can substitute this value back into one of the original equations to find \( y \). Let's use the first equation:

\[
y = 23\left(\frac{1}{6}\right) - 3
\]

Calculating:

\[
y = \frac{23}{6} - 3 = \frac{23}{6} - \frac{18}{6} = \frac{5}{6}
\]

So the solution to the system of equations is:

\[
\left(\frac{1}{6}, \frac{5}{6}\right)
\]

Now, let's look at the responses provided:
- There are infinite solutions.
- (6,1)
- There is no solution.
- (0,3)

None of these options match the solution we determined, \(\left(\frac{1}{6}, \frac{5}{6}\right)\). Therefore, it looks like the provided options are incorrect for this system of equations.