Linear Functions: ax+by=c

The general form for a line is

    \[ax+by=c\]

where a, b and c are integers.

How does it work? Video

Using either the slope intercept form (y=mx+b) or the general form (ax+by=c) we can calculate all the critical values of a straight line. These are: the x intercept, the y intercept and the slope.

Example 1: convert 3x+2y=-5 to y=mx+b form:

    \begin{align*}3x+2y&=-5 &\text{take $3x$ from both sides} \\[10pt] 2y&=-3x-5&\text{divide all terms by 2}\\[10pt] y&=-\frac{3}{2}x-\frac{5}{2}\end{align*}

Example 2: convert y=\frac{1}{4}x+3 to general form:

    \begin{align*} y&=\frac{1}{4}x+3 &\text{multiply every term by 4}\\[10pt] 4y& = x + 12&\text{subtract 12 from both sides}\\[10pt] 4y-12&=x &\text{subtract $4y$ from both sides}\\[10pt] -12&=x-4y\end{align*}

Finding the y intercept

The y intercept is the point where the line cuts the y-axis. The x coordinate is zero. To find the y intercept, we substitute 0 for x.

Example 3: Find the y intercept of the line 3x+2y=-5

    \begin{align*}3x+2y&=-5 &\text{substitute $x=0$} \\[10pt] 3(0)+2y&=-5&\text{simplify}\\[10pt] 2y&=-5 &\text{divide both sides by 2}\\[10pt] y&=-\frac{5}{2}&\text{$y$ intercept is  }-\frac{5}{2}\end{align*}

Finding the x intercept

The x intercept is the point that is on the line and on the x axis. The y coordinate is zero. To find the x intercept, we substitute 0 for y.

Example 4: Find the x intercept of the line 3x+2y=-5

    \begin{align*}3x+2y&=-5 &\text{substitute $y=0$} \\[10pt] 3x+2(0)&=-5&\text{simplify}\\[10pt] 3x&=-5 &\text{divide both sides by 3}\\[10pt] x&=-\frac{3}{2}&\text{$x$ intercept is  }-\frac{3}{2}\end{align*}

Calculations

Example: Video (3:31 mins)

5 Examples for this applet: Video (16 mins)

Examples for this applet on a pdf: Examples x y intercepts slope general form

Calculate the x and y intercepts and the slope. Rewrite the equation in the alternate form. Use fractions or terminating decimals, don’t round decimals.


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