Linear Functions: ax+by+c=0

Udothemath printable: 10 FMP Unit 6 Skill 5

The general form (Skill 5)

The general form for a line is

    \[ax+by+c=0\]

where a, b and c are integers. One may rearrange this form to the more familiar slope, y intercept form y=mx+b.

Example 1: convert 3x+2y+5=0 to y=mx+b form:

    \begin{align*}3x+2y+5&=0 \\ 2y&=-3x-5&\text{take 3x and take 5 from both sides}\\ y&=-\frac{3}{2}x-\frac{5}{2}&\text{divide all terms by 2}\end{align*}

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

    \begin{align*} y&=\frac{1}{4}x+3 \\ 4y& = x + 12&\text{multiply every term by 4}\\ 0&=x+12-4y&\text{subtract 4y from both sides} \\ x-4y+12&=0&\text{swap sides and reorder the terms}\end{align*}

On this applet, find the general form for each line and type it in the box. If correct, the green line will match the blue line. Any correct equivalent expression will work.

Applet 1

Finding the y intercept

The y intercept is the point that is on the line and on 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=0

Solution: 3(0)+2y+5=0

Leads us to 2y+5=0

Solving we have 2y=-5 and therefore, y=-\frac{5}{2}

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=0

Solution: 3x+2(0)+5=0

Leads us to 3x+5=0

Solving we have 3x=-5 and therefore, x=-\frac{5}{3}

Calculations (Skill #5)

Use the applet to practise converting to the form y=mx+b, and calculating the x and y intercepts.

Applet 2


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