The graphs represent this system:
y=3x−2
−3x+y=7
How many solutions does this system have: 0, 1, or ∞ ?
The two equations can be represented by the following system:
1) y = 3x - 2
2) -3x + y = 7
We can rewrite equation 2 in terms of y:
2) y = 3x + 7
The two equations are represented by two lines on a coordinate plane. The first equation has a slope of 3 and a y-intercept of -2. The second equation also has a slope of 3 but a y-intercept of 7.
Since the slopes are the same, the lines are parallel and will never intersect. Therefore, there are no solutions to this system.
The answer is 0.
To determine the number of solutions, let's solve the system of equations.
First, we can rewrite the second equation in terms of y:
-3x + y = 7
y = 3x + 7
Now that we have both equations in slope-intercept form, we can see that they have the same slope (3). The line represented by the equation y = 3x - 2 has a y-intercept of -2, while the line represented by y = 3x + 7 has a y-intercept of 7.
Since these lines have different y-intercepts but the same slope, they are parallel. When two lines are parallel, they never intersect, indicating that there are no solutions for this system of equations.
Therefore, the system has 0 solutions.
To determine the number of solutions for this system of equations, we can examine the lines represented by the given equations. The first equation, y = 3x - 2, is in slope-intercept form (y = mx + b), where m is the slope (3) and b is the y-intercept (-2).
To graph the first equation, we can start by plotting the y-intercept, which is the point (0, -2). From this point, we can use the slope (3) to find additional points. To find the next point, we can move 1 unit in the positive x-direction (to the right) and then move 3 units in the positive y-direction (upward). Connecting these points will give us a straight line.
Now, let's examine the second equation: -3x + y = 7. To graph this equation, we can rearrange it to slope-intercept form by isolating y:
-3x + y = 7
y = 3x + 7
Comparing this equation to the first one, we can see that they have the same slope (3) but different y-intercepts (-2 and 7).
Now, since both lines have the same slope, they are parallel and will never intersect. In other words, there are no solutions that satisfy both equations simultaneously. Therefore, this system of equations has no solutions.
In conclusion, the number of solutions for this system is 0.