# Points and Lines on the ACT and SAT

Given the coordinates of two points, you can find out many things such as

- The slope or gradient (these two words mean the same thing) of the line connecting the two points
- The equation of the line connecting the two points
- Correspondingly, the x and y-intercepts of said line
- The slope of another line that is perpendicular to the first line.

### Slope of a line

Suppose we have two points with coordinates (3,8 ) and (-2, 2). If there is a line connecting these two points, how do we find the slope of this line?

- Subtract the y-coordinate of the second point from the first point: 8 – 2 in this case. i.e. 6
- Subtract the x-coordinate of the second point from the first point: 3 – (-2) in this case i.e. 5
- Divide your answer in step 1 by your answer in step 2: 6 / 5 = 1.2

In general, if you have two points with coordinates (x_{1}, y_{1}) and (x_{2}, y_{2}) then the formula for the slope is: y_{1}– y_{2/}x_{1 – }x_{2}

### Equation of a line

The general formula for a line is

*y*= m

*x*+ c where m is the slope of the line and c is the y-intercept (where the line cuts the y-axis). We have already learned from the previous part how to find the slope of a line given 2 points on the line. Here’s how we find the equation of the line.

- Take a point with coordinates x and y e.g. the point (3,8 )) from the previous example
- Take y –
*y-coordinate*i.e. y – 8 - Take x –
*x-coordinate*i.e. x – 3 - Multiply Step 3 by the slope, which we have found to be 1.2 earlier. We get 1.2 (x – 3)
- Equate Step 2 to Step 4. i.e. y – 8 = 1.2 (x – 3)
- The equation in Step 5 is the equation of the line. Simplify it to get y = 1.2 x + 4.4

In general, to get the equation of a line from two points, find the slope first (call it *m*) then plug in the values of one point into the following equation to get the equation of the line

y – y_{1 – m(}x –x1) where m is given by the formula for slope: y_{1}– y_{2/}x_{1 – }x_{2}

#### Intercepts

We can directly read off what the y-intercept is from the equation of the line. From the previous example, we know that the line intersects the y-axis at (0, 4.4). To find the x-intercept, we substitute y=0 into the equation. We then get 0 = 1.2x + 4.4 Rearranging that quickly, we realize that x = -4.4 / 1.2 which is approximately -3.67. So the line intersects the x-axis at (0, -3.67)

In general, if we have a line

*L*and another line

_{1}*L*with slopes

_{2}*m*and

_{1 }*m*then the two lines are related by the following formula

_{2}(

*m*) (

_{1}*m*) = -1

_{2}So for the line we found in the example above, the slope of any line perpendicular to it is -1/1.2 which is -5/6 or approximately 0.83

Note that there are infinite number of lines that are perpendicular to the original line example y=1.2x + 4.4 If you are given an addition point that lines on the new perpendicular line, you will be able to find the exact equation of the perpendicular line.