Linear Function Equation

The x and y coordinates of points on a straight line have a numerical pattern. There are three numerical patterns to observe on this line:

Remember that the first number in a coordinate pair is for x, the second is for y, that is: (x,y).

  • The points are plotted such that the x coordinates are going up in ones;
  • The y coordinates are going up in 4s;

More importantly, there is a pattern between the two numbers in each ordered pair. That is, the y coordinate is four times the x coordinate then add one.

Take the point (2,9). Notice, 4 \times 2 +1=9.

Also, (5,21). Notice, 4 \times 5 +1 = 21.

This calculation works for all the coordinate pairs on the line.

We can capture this calculation with an equation:

    \[y=4x+1\]

Notice that the number we multiply x by is 4. This is the same number the y coordinates are going up in (when the x coordinates go up in ones). We use the letter m to represent this number (the multiplier); and we call it the slope because the multiplier determines how steep the line is.

As long as we know two points on a line, we can calculate the slope and then figure out the equation of the line.

To calculate slope, m, we can use:

    \[m=\frac{\text{rise}}{\text{run}}\]

or the formula

    \[m=\frac{y_2-y_1}{x_2-x_1}\]

Once we know the slope, we know that the equation of the line begins

    \[y=mx + \dots\]

If one of the points we know lies on the y axis, that is, the point has x=0, we call it the y intercept of the line. This is the number that we complete the equation with.

Example: Determine the equation of the line that goes through the points (0,7),(1,10),(2,13),(5,22),(10,37)

We only need to use two points to find the equation. Let’s use (0,7) because this is the y intercept and (10,37).

    \[m=\frac{y_2-y_1}{x_2-x_1}=\frac{37-7}{10-0}=\frac{30}{10}=3\]

That tells us that m=3 and so our equation begins y=3x + \dots.

The y intercept has coordinates (0,7) and so our equation finishes with +7

    \[y=3x+7\]

We check with some other coordinate pairs:

Use point (3,16): y=3(3)+7=9+7=16 \checkmark

Use point (5,22): y=3(5)+7=15+7=22 \checkmark.

Practice:

GeoGebra applet link

Test out on Slope y intercept


Any Two Points

To find the equation of a line using any two points, we begin by calculating the slope with the formula

    \[m=\frac{y_2-y_1}{x_2-x_1}\]

We then substitute the x and y coordinates of one known point (x_1,y_1) and the slope m into the equation

    \[y_1=m\,x_1+b\]

to calculate the value b.

Example: Find the equation of the line that goes through the points (-1,20), (4,10), and (7,4).

Solution: We need only two points to calculate the equation. Let’s use (-1,20) and (7,4).

First, calculate slope:

    \[m=\frac{y_2-y_1}{x_2-x_1}=\frac{4-20}{7-(-1)}=\frac{-16}{8}=-2\]

Second, calculate b using slope m=-2, and any point. Let’s use (7,4).

    \begin{align*}y_1&=m\,x_1+b\\[10 pt]4&=-2(7)+b\\[10 pt]4&=-14+b\\[10 pt]18&=b\end{align}

We now have m=-2 and b=18, therefore the equation of the line is

    \[y=-2x+18\]

which could also be written as

    \[y=18-2x\]

just by changing the order of the terms.

Practice:

GeoGebra applet link


A more efficient formula

If we streamline the process of calculating b, we arrive at the formula

    \[y-y_1=m(x-x_1)\]

This is sometimes known as the ‘point/slope’ formula.

Using the same example as before, calculate the equation of the line that goes through the points (-1,20) and (7,4)

We need to calculate the slope m in the same way:

    \[m=\frac{y_2-y_1}{x_2-x_1}=\frac{4-20}{7-(-1)}=\frac{-16}{8}=-2\]

However, to calculate the equation we substitute m=-2 and choose any known point for the (x_1,y_1). Let’s use (7,4) again:

    \begin{align*}y-y_1&=m(x-x_1)\\[10 pt]y-4&=-2(x-7)\\[10 pt]y-4&=-2x+14\\[10 pt]y&=-2x+18\end{align}

Why it works

This formula y-y_1=m(x-x_1) is derived as follows:

    \begin{align*}y_1&=m\,x_1+b\\[10pt]y_1-m\,x_1&=b\end{align}

Now substitute y_1-mx_1 in place of b in the formula y=mx+b and rearrange:

    \begin{align*}y&=mx+b\\[10 pt]y&=mx+y_1-m\,x_1\\[10 pt]y-y_1&=mx-mx_1\\[10 pt]y-y_1&=m(x-x_1)\quad\text{as required.}\end{align}


Use any method to calculate the equation given the coordinates of one point and the slope:

Printed practice:Grade 10 Point Slope

GeoGebra applet link


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