Question

y=3x−2

−3x+y=7
 

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

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Answer 1

[AI-generated answer — Step-by-Step Bot]

To determine the number of solutions, we need to solve the system of equations. Let's solve the system using the method of substitution.

We will start by solving one of the equations for one variable and then substitute that expression into the other equation.

Given:
1) y = 3x - 2
2) -3x + y = 7

We can rearrange equation 2 to solve for y:
-3x + y = 7
y = 3x + 7

Now we can substitute this expression for y into equation 1:

3x + 7 = 3x - 2

By subtracting 3x from both sides, we get:

7 = -2

Since we have a contradiction (7 = -2 is clearly false), this system of equations has no solution.

Therefore, the number of solutions for this system is 0.

Answer 2

[AI-generated answer — Explain Bot]

To determine the number of solutions of a system of equations, you can use the concept of slopes. If the slopes of the two lines are different, then the lines will intersect at one point, and the system has one solution. If the slopes are equal and the y-intercepts are different, then the lines will never intersect, and the system has no solution. If the slopes are equal and the y-intercepts are also equal, the lines will coincide, and the system has infinite solutions.

Let's start by putting both equations into slope-intercept form, which is in the form y = mx + b, where m is the slope and b is the y-intercept.

First equation: y = 3x - 2
Second equation: -3x + y = 7

To put the second equation into slope-intercept form, we need to solve it for y:
y = 3x + 7

Comparing the slopes of both equations, we can see that the slopes are not equal (3 and 3). Therefore, the lines will intersect at one point, and the system has one solution.

Therefore, the system of equations has 1 solution.