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)
This system of equations has 1 solution.
To determine the number of solutions for the system represented by the given equations, we need to find the point(s) where the solid line and the dashed line intersect.
First, let's solve the system of equations to find the intersection point:
1. Equation 1: y = 3x - 2
2. Equation 2: -3x + y = 7
To find the intersection point, we can substitute the value of y from Equation 1 into Equation 2:
-3x + (3x - 2) = 7
Simplifying, we get:
-3x + 3x - 2 = 7
-2 = 7
Since -2 does not equal 7, this system of equations has no solution.
Therefore, the answer is 0 solutions.
To determine the number of solutions that this system of equations has, we need to analyze the lines represented by the equations on the coordinate plane.
The solid line has an equation y = 3x - 2. This equation represents a line with a slope of 3 and a y-intercept of -2. The line is increasing from left to right since the coefficient of x is positive.
The dashed line has an equation -3x + y = 7. To find the slope-intercept form of this equation, we need to isolate y. Adding 3x to both sides of the equation gives us y = 3x + 7, which is the same as y = 3x + 7. This equation represents a line with the same slope of 3 as the solid line, but it has a different y-intercept of 7.
Since both lines have the same slope, they are parallel. Parallel lines never intersect, which means there is no point of intersection, and therefore, no solution to this system of equations.
In conclusion, this system has 0 solutions.