Question

How do you tell whether the system `y= -2x + 1` and `y = -1/3x - 3`has no solution or infinitely many solutions?

Answer 1

If you were to try to find the solution(s) graphically, you would plot both of the equations as straight lines. The solution(s) are where the lines intersect. As these are both straight lines, there would be, at most, one solution. Since the lines are not parallel (the gradients are different), you know that there is a solution. You can find this graphically as just described, or algebraically.

`y=-2x+1` and `y=-1/3x-3`
So
`-2x+1=-1/3x-3`
`1=5/3x-3`
`4=5/3 x`
`x=12/5=2.4`

Answer 2

See explanation.

Explanation:

`color(blue)("Answering the question as stated")`

The first condition for either no solution or an infinite count of solutions is that they must be parallel.

No solution parallel and different y or x intercepts

Infinite solutions parallel and the same y or x intercept
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Investigating the given equations")`

Given:
`y=-2x+1`
`y=-1/3x-3`

`color(brown)("Are they parallel? No!")`

The values in front of the `x` (coefficients) determine the slope. As they are different values the slopes are different so it is not possible for them to be parallel.

`color(brown)("Do they have the same y-intercept? No!")`

`color(green) (y=-2xcolor(red)(+1)`
`color(green)(y=-1/3xcolor(red)(-3))`

The red constants at the end are the y-intercepts and they are of different value

`color(brown)("Where do they cross each other?")`

`color(brown)("Not going to do the maths but I will show you the graph")`