7.4 The slope-intercept form of a line

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$\begin{array}{l} y = 7 x . & In this case m = 7 and b = 0 since we can write y = 7 x as y = 7 x + 0. \end{array}$

The following equations are not in slope-intercept form:

$\begin{array}{l} 2 y = 4 x - 1. & The coefficient of y is 2. \end{array} To be in slope-intercept form, the coefficient of y must be 1.$

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$\begin{array}{l} y + 4 x = 5. & The equation is not solved for y . \end{array} The x and y appear on the same side of the equal sign .$

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$\begin{array}{l} y + 1 = 2 x . & The equation is not solved for y . \end{array}$

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Practice set a

The following equation are in slope-intercept form. In each case, specify the slope and $y -intercept$ .

$\begin{matrix} y = 2 x + 7; & m = & b = \end{matrix}$

$m = 2, b = 7$

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$\begin{matrix} y = - 4 x + 2; & m = & b = \end{matrix}$

$m = - 4, b = 2$

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$\begin{matrix} y = - 5 x - 1; & m = & b = \end{matrix}$

$m = - 5, b = - 1$

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$\begin{matrix} y = \frac{2}{3} x - 10; & m = & b = \end{matrix}$

$m = \frac{2}{3}, b = - 10$

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$\begin{matrix} y = \frac{- 5}{8} x + \frac{1}{2}; & m = & b = \end{matrix}$

$m = \frac{- 5}{8}, b = \frac{1}{2}$

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$\begin{matrix} y = - 3 x; & m = & b = \end{matrix}$

$m = - 3, b = 0$

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Slope and intercept

When the equation of a line is written in slope-intercept form, two important properties of the line can be seen: the slope and the intercept . Let's look at these two properties by graphing several lines and observing them carefully.

Sample set b

Graph the line $y = x - 3$ .

$x$	$y$	$(x, y)$
0	$- 3$	$(0, - 3)$
4	1	$(4, 1)$
$- 2$	$- 5$	$(- 2, - 5)$

A graph of a line passing through three points with coordinates zero, negative three; four, one , and negative two, negative five.

Looking carefully at this line, answer the following two questions.

At what number does this line cross the $y -axis$ ? Do you see this number in the equation?

The line crosses the $y -axis$ at $- 3$ .

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Place your pencil at any point on the line. Move your pencil exactly one unit horizontally to the right. Now, how many units straight up or down must you move your pencil to get back on the line? Do you see this number in the equation?

After moving horizontally one unit to the right, we must move exactly one vertical unit up. This number is the coefficient of $x$ .

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Graph the line $y = \frac{2}{3} x + 1$ .

$x$	$y$	$(x, y)$
0	1	$(0, 1)$
3	3	$(3, 3)$
$- 3$	$- 1$	$(- 3, - 1)$

A graph of a line passing through three points with coordinates zero, one; three, three, and negative three, negative one.

Looking carefully at this line, answer the following two questions.

At what number does this line cross the $y -axis$ ? Do you see this number in the equation?

The line crosses the $y -axis$ at $+ 1$ .

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After moving horizontally one unit to the right, we must move exactly $\frac{2}{3}$ unit upward. This number is the coefficient of $x$ .

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Practice set b

Graph the line $y = - 3 x + 4$ .

$x$	$y$	$(x, y)$
0
3
2

An xy coordinate plane with gridlines, labeled negative five and five on the both axes.

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Looking carefully at this line, answer the following two questions.

At what number does the line cross the $y -axis$ ? Do you see this number in the equation?

The line crosses the $y -axis$ at $+ 4$ . After moving horizontally 1 unit to the right, we must move exactly 3 units downward.

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A graph of a line passing through two points with coordinates zero,four; two, negative two ,and three, negative five.

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In the graphs constructed in Sample Set B and Practice Set B, each equation had the form $y = m x + b$ . We can answer the same questions by using this form of the equation (shown in the diagram).

A graph of a line sloped up and to the right in a first quadrant labeled with the equation y equal to mx plus b and intersecting y axis at point with coordinates zero, b. Lines illustrating an upward change of m units and a horizontal change of one unit to the right.

$y$ -intercept

At what number does the line cross the $y -axis$ ? Do you see this number in the equation?

In each case, the line crosses the $y -axis$ at the constant $b$ . The number $b$ is the number at which the line crosses the $y -axis$ , and it is called the $y -intercept$ . The ordered pair corresponding to the $y -intercept$ is $(0, b) .$

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To get back on the line, we must move our pencil exactly $m$ vertical units.

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Source: OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3

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