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Linear Equations and Graphs

Linear Equations

The linear function y = mx + b can be re-written in many forms, for example –mx + y = b. This is a special case of the linear equation ax + by = c.

The points satisfying a linear equation, no matter which form it is written in, form a line. We consider two cases of the linear equation ax + by = c.

  • Case 1: b 0.
    In this case, we can rewrite the equation as y=(-a/b)*x + c/b .

    This gives a line with slope m=-a/b and y-intercept = c/b.
  • The x-intercept is the point at which the line crosses the x-axis. Along the x-axis, y = 0. Solving the linear equation for x when y = 0 gives

                   0=(-a/b)*x + c/b which implies x=c/a.

    and the x-intercept is therefore c/a.


    For example, the slope of the linear equation 3x + 4y = 12 is -3/4, the y-intercept is 12/4=3 and the x-intercept is 12/3 = 4.
  • Case 2: b =0, a 0.
    In this case the equation becomes ax = c, hence   x=c/a   regardless of what y is. This is a vertical line passing through x=c/a .

As mentioned before under the topic of the slope, the slope of a vertical line is undefined.
It is not the graph of a function.

Exercise 8.

Complete the following table.

(If the slope is undefined, type DNE in the slope box (for does not exist). Remember that if the slope is undefined, the line is vertical and there is an x-intercept, but there is no unique y-intercept.
If the slope is undefined, you cannot enter a function of the graph in the table below.
Round any answers to two decimal places.)

Equation
Slope
x-intercept
y-intercept
Function

x + y =

( , )
( , )
y = x +

 

<< Determining Linear Equations (or Functions) from Info. about the Line | Linear Equations and Graphs Index | Parallel and Perpendicular Lines >>

 

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