Using Standard Form

The standard form of writing a line is $Ax+By=C,$ where (A,) $B,$ and $C$ are integers.

This form is particularly useful for determining both the $x$- and $y$-intercepts of a line. We can determine the $x$-intercept by substituting 0 for $y$ and solving for $x.$ Similarly, we can determine the $y$-intercept of the line by substituting 0 for $x$ and solving for $y.$

If the equation of a line is $3x + 5y = 60,$ what are the $x$-intercept and $y$-intercept of the line?

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ANSWER

To find the $x$-intercept, we substitute 0 for $y$ and solve: $\begin{aligned} 3x + 5(0) &= 60 \\ 3x &= 60 \\ x &= 20.\end{aligned}$

To find the $y$-intercept, we substitute 0 for $x$ and solve: $\begin{aligned} 3(0) + 5y &= 60 \\ 5y &= 60 \\ x &= 12.\end{aligned}$

The $x$-intercept is $(12,0)$ and the $y$-intercept is $(0,20).$

If the $x$-intercept and $y$-intercept of a line are $(5,0)$ and $(0,6)$, respectively, what is the equation of the line?

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ANSWER

Dividing both sides of the standard form equation by $C$ yields the equation $\frac{A}{C}x+\frac{B}{C}y=1.$ Given this equation, the $x$-intercept is $\left(\frac{C}{A},0\right)$ and the $y$-intercept is $\left(0,\frac{B}{C}\right).$

Since our $x$-intercept is 5, $\frac{A}{C} = \frac{1}{5}.$ Since our $y$-intercept is 6, $\frac{B}{C} = \frac{1}{6}.$

Substituting our known values into the equation, we have $\frac{1}{5}x + \frac{1}{6}y = 1.$ Multiplying both sides by $30$ yields $6x + 5y = 30$. $_\square$