Use the image to answer the 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)
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The system of equations represented by the solid line (y=3x-2) and the dashed line (-3x+y=7) has 1 solution. This is because the two lines intersect at exactly one point on the coordinate plane.
To determine the number of solutions for the system of equations represented by the graphs, we need to see if the two lines intersect.
First, let's focus on the equation of the solid line: y=3x−2.
Now, rearranging the equation of the dashed line −3x+y=7, we get:
y=3x+7.
Now, we can see that the two equations have the same slope, which is 3. Since the slopes are the same and the y-intercepts are different, the lines are parallel and will never intersect.
Therefore, the system has no solutions.
To determine the number of solutions for this system of equations, we can visually analyze the graph.
The solid line represents the equation y = 3x - 2, and the dashed line represents the equation -3x + y = 7.
The equation y = 3x - 2 is in slope-intercept form, with a slope of 3 and a y-intercept of -2. This line will have a positive slope and will go through the point (0, -2).
The equation -3x + y = 7 can be rearranged to y = 3x + 7, which is also a linear equation with the same slope as the solid line but a different y-intercept. This line will go through the point (0, 7).
Since both lines have the same slope, they are parallel to each other and will never intersect. Therefore, the system of equations represented by these lines has no solutions, because there are no points that satisfy both equations simultaneously.
So the answer to the question "How many solutions does this system have?" is 0.