Linear Functions: ax+by+c=0

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*}

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=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)

Calculate the x and y intercepts. Rewrite the equation in the form y=mx+b. Use an integer, a terminating decimal or a fraction for the coefficient of x (that is, don’t round a decimal).


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