Use the image to answer the question.

Question

A coordinate plane with 4 quadrants ranges from negative 10 to 10 in unit increments on both the x and y axes. A solid line and a dashed line with arrows at both the ends are drawn parallel to each other on the graph. The equation of the solid line is y equals 3 x minus 2. The equation of the dashed line is negative 3 x plus y equals 7.

The graphs represent this system:  

y=3x−2


−3x+y=7
 

How many solutions does this system have: 0, 1, or ∞
?

(1 point)

Answer 1

To determine the number of solutions for the system of equations given, we need to analyze the lines represented by their equations.

1. The first equation is \(y = 3x - 2\). This is the equation of a solid line with a slope of 3.

2. The second equation is \(-3x + y = 7\). To write it in slope-intercept form (y = mx + b), we can rearrange it: \[ y = 3x + 7 \] This is the equation of a dashed line with a slope of 3.

Since both lines have the same slope (3), they are parallel. However, a solid line usually represents a line that includes all points on it, while a dashed line usually represents a line that does not include the points on it.

Additionally, the solid line and the dashed line have different y-intercepts (the solid line has a y-intercept of -2 and the dashed line has a y-intercept of 7), indicating that they never intersect.

Since the two lines are parallel and will never meet, the system of equations has no solutions.

The answer is \(0\).