Question
he following system of linear equations is shown in the graph.
y equals one third times x plus 1
x + 3y = −3
a coordinate plane with one line that passes through the points 0 comma 1 and 3 comma 2 and another line that passes through the points 0 comma negative 1 and 3 comma negative 2
How many solutions does the system of linear equations have?
No solution
Infinitely many solutions
One solution at (−3, 0)
One solution at (3, −2)
y equals one third times x plus 1
x + 3y = −3
a coordinate plane with one line that passes through the points 0 comma 1 and 3 comma 2 and another line that passes through the points 0 comma negative 1 and 3 comma negative 2
How many solutions does the system of linear equations have?
No solution
Infinitely many solutions
One solution at (−3, 0)
One solution at (3, −2)
Answers
GPT-4o mini
To determine how many solutions the given system of linear equations has, we need to understand the two equations presented:
1. The first equation is given in slope-intercept form:
\[
y = \frac{1}{3}x + 1
\]
This line passes through the points (0, 1) and (3, 2).
2. The second equation is in standard form:
\[
x + 3y = -3
\]
Let's rewrite it in slope-intercept form:
\[
3y = -x - 3 \\
y = -\frac{1}{3}x - 1
\]
This line passes through the points (0, -1) and (3, -2).
### Analyzing the Graph
From the equations:
- The first line has a slope of \(\frac{1}{3}\) (positive slope).
- The second line has a slope of \(-\frac{1}{3}\) (negative slope).
Since the slopes are different, the two lines intersect at exactly one point, which means there is **exactly one solution** to the system of equations.
### Conclusion
So, the correct answer to the number of solutions for the system of linear equations is:
**One solution.**
It appears that while potential intersection points are mentioned, the correct unique solution is not explicitly listed in the provided options. However, it is still conclusive that the two lines intersect at exactly one point.
1. The first equation is given in slope-intercept form:
\[
y = \frac{1}{3}x + 1
\]
This line passes through the points (0, 1) and (3, 2).
2. The second equation is in standard form:
\[
x + 3y = -3
\]
Let's rewrite it in slope-intercept form:
\[
3y = -x - 3 \\
y = -\frac{1}{3}x - 1
\]
This line passes through the points (0, -1) and (3, -2).
### Analyzing the Graph
From the equations:
- The first line has a slope of \(\frac{1}{3}\) (positive slope).
- The second line has a slope of \(-\frac{1}{3}\) (negative slope).
Since the slopes are different, the two lines intersect at exactly one point, which means there is **exactly one solution** to the system of equations.
### Conclusion
So, the correct answer to the number of solutions for the system of linear equations is:
**One solution.**
It appears that while potential intersection points are mentioned, the correct unique solution is not explicitly listed in the provided options. However, it is still conclusive that the two lines intersect at exactly one point.