Forms of Linear Equations

The three major forms of linear equations are slope-intercept form, point-slope form, and standard form.

The slope-intercept form is $$y = mx + b.$$

It provides us with two important pieces of information about the graph of a line: the slope $m$ and the $y$-intercept $(0, b )$.

What is the slope of $y = 2x +3$?

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ANSWER

Since the slope is the value of $m$, we can see that the slope of this equation is $2.\ _\square$

What is the $y$-intercept of $y = -4x +10$?

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ANSWER

Since the $y$-intercept occurs when $x = 0$, we can see that the $y$-intercept is $10.\ _\square$

We use point-slope form when we know a point $(x_1, y_1)$ on the line, and the slope $m$. Given this information, the equation of the line is

$$( y - y_1) = m ( x - x_1).$$

Find the equation of the straight line that passes through the point $(3, 5)$ and has slope $- 2$.

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ANSWER

From the point-slope form, the equation is $( y - 5) = -2 ( x - 3)$, which can be simplified as

$$y = -2x + 11 . \ _\square$$

Find the equation of the line which passes through the points $P = (1, 7)$ and $Q = ( -1 , -3 )$.

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ANSWER

The slope of the line is $m = \frac{ 7 - ( -3) } { 1 - (-1) } = \frac{ 10} { 2} = 5$. Hence, the equation of the line is $(y - 7) = 5 ( x - 1)$, or $y = 5x +2$. $_\square$

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

This form of a line is particularly useful for determining both the $x$- and $y$-intercepts. We can determine the $x$-intercept of the line by substituting 0 for $y$ and solving for $x.$ 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: $$\begin{aligned} 3(0) + 5y &= 60 \\ 5y &= 60 \\ y &= 12.\end{aligned}$$

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

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 $\left(0,\frac{C}{B}\right).$

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 $6x + 5y = 30$. $_\square$