Question

What is the slope of a line that passes through `(-2, -3)` and `(1, 1)`?

Answer 1

Use the two coordinates formula to figure out the equation of a straight line.

Explanation:

I do not know if by slope you mean the equation of the line or simply the gradient.

Gradient Only Method

To get the gradient you simply do `dy/dx` which means difference in `y` over difference in `x`

The formula expanded means we do `(y_2-y_1)/(x_2-x_1)` where our coordinates are `(x_1,y_1)` and `(x_2,y_2)`

For your example we substitue the values in to get `(1-(-3))/(1-(-2))`

This turns into `(1+3)/(1+2)` simplified this is `4/3` so your gradient or 'slope' is `4/3` or `1.dot 3`

Equation of Straight Line Method

As for the full equation we use the two coordinates formula.

This formula is: `(y-y_1)/(y_2-y_1) = (x-x_1)/(x_2-x_1)` where our coordinates are `(x_1,y_1)` and `(x_2,y_2)`.

If we substitute in your values we get: `(y-(-3))/(1-(-3)) = (x-(-2))/(1-(-2))`

Tidying up the negatives we get: `(y+3)/(1+3) = (x+2)/(1+2)`

Simplifying we get: `(y+3)/4 = (x+2)/3`

Now we must rearrange this expression into the form `y=mx+c`

To do this we will first multiply both sides by 4 to remove the fraction. If we do this we get: `y+3 = (4x+8)/3`

Then we will multiply both sides by 3 to remove the other fraction. This gives us: `3y+9 = 4x+8`

Take away 9 from both sides to get y on its own: `3y = 4x-1`

Then divide by 3: `y = 4/3x - 1/3`

In this case you can also get the gradient as the `m` part of the equation: `y=mx+c` is the gradient. Which means that the gradient is `4/3` or `1.dot 3` as we got using the first method.

Interestingly we can also use the `c` part of the equation to figure out the `y` intercept. In this case it is `1/3` which means the `y` intercept of this line is at the coordinate `(1/3,0)`