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Writing linear equations using the slope-intercept form

An equation in the slope-intercept form is written as

\\ y=mx+b \\

Where m is the slope of the line and b is the y-intercept. You can use this equation to write an equation if you know the slope and the y-intercept.

Example:

Find the equation of the line

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Choose two points that are on the line

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Calculate the slope between the two points

\\ m=\frac{y_{2}\, -y_{1}}{x_{2}\, -x_{1}}=\frac{\left (-1 \right )-3}{3-\left ( -3 \right )}=\frac{-4}{6}=\frac{-2}{3} \\

We can find the b-value, the y-intercept, by looking at the graph

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b = 1

We've got a value for m and a value for b. This gives us the linear function

\\ y=-\frac{2}{3}x+1 \\

In many cases the value of b is not as easily read. In those cases, or if you're uncertain whether the line actually crosses the y-axis in this particular point you can calculate b by solving the equation for b and then substituting x and y with one of your two points.

We can use the example above to illustrate this. We've got the two points (-3, 3) and (3, -1). From these two points we calculated the slope

\\ m=-\frac{2}{3} \\

This gives us the equation

\\ y=-\frac{2}{3}x+b \\

From this we can solve the equation for b

\\ b=y+\frac{2}{3}x \\

And if we put in the values from our first point (-3, 3) we get

\\ b=3+\frac{2}{3}\cdot \left ( -3 \right )=3+\left ( -2 \right )=1 \\

If we put in this value for b in the equation we get

\\ y=-\frac{2}{3}x+1 \\

which is the same equation as we got when we read the y-intercept from the graph.

To summarize how to write a linear equation using the slope-interception form you

1.       Identify the slope, m. This can be done by calculating the slope between two known points of the line using the slope formula.
2.       Find the y-intercept. This can be done by substituting the slope and the coordinates of a point (x, y) on the line in the slope-intercept formula and then solve for b.

Once you've got both m and b you can just put them in the equation at their respective position.

Videolesson: Find the equation to the graph

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Next Class:  Formulating linear equations, Writing linear equations using the point-slope form and the standard form