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 answer is 1. The two lines intersect at a single point, indicating that there is one solution to the system of equations.
are you sure
I apologize for the confusion. Upon closer examination of the given information, it appears that the two lines are parallel to each other. Since parallel lines do not intersect, the system of equations has no solution. Therefore, the answer is 0.
To determine the number of solutions in this system, let's analyze the two equations separately:
1. The equation of the solid line is y = 3x - 2.
This equation represents a straight line with a slope of 3 and y-intercept of -2.
2. The equation of the dashed line is -3x + y = 7.
Rearranging this equation, we get y = 3x + 7.
This equation also represents a straight line with a slope of 3 and y-intercept of 7.
Since both lines have the same slope (3), but different y-intercepts, they are parallel.
Two parallel lines on the coordinate plane will either have 0 solutions (if they do not intersect) or infinite solutions (if they are the same line).
In this case, since the lines have different y-intercepts, they will never intersect. Therefore, the answer is 0 solutions.
So, the system represented by these equations has 0 solutions.
To determine the number of solutions for the given system of equations, we need to analyze the relationship between the two lines represented by the equations.
First, let's consider the equation y = 3x - 2, which is represented by a solid line on the graph. This equation represents a linear relationship where the slope is 3 (the coefficient of x) and the y-intercept is -2 (the constant term).
Next, let's analyze the equation -3x + y = 7, which is represented by a dashed line on the graph. To write it in slope-intercept form, let's isolate y:
y = 3x + 7.
Comparing this equation to the solid line equation, we can see that both lines have the same slope but different y-intercepts. Therefore, the two lines are parallel.
When two parallel lines are graphed, they will never intersect. In the context of solving a system of equations, this means that there are no solutions. Hence, the system represented by y = 3x - 2 and -3x + y = 7 has 0 solutions.
So, the answer to the question is 0 solutions.