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 « Basic Concepts
Analytic Geometry: (lesson 2 of 4)

## Lines

Only two pieces of information are needed to completely describe a given line. However, there is some flexibility on which two pieces of information are used:

1. specifying the slope and the "y intercept", $b$, of the line (slope - intercept form).

2. specifying the slope of the line and one point on the line (point slope form).

3. specifying two points through which the line passes (two point form).

### Slope-Intercept Form

The most useful form of straight-line equations is the "slope-intercept" form:

$y = mx + b$

This is called the slope-intercept form because $m$ is the slope and $b$ gives the y-intercept. That means the point $(0, b)$ is where the line crosses the y-axis.

The Slope-Intercept Form of the equation of a straight line introduces a new concept, that of the y-intercept. The y-intercept describes the point where the line crosses the y-axis. At this set of coordinates, the $y$ value is zero, and the $x$ value is the y-intercept.

Example 1:

1. $y = 5x + 7$

2. $y = -3x + 23$

3. $y = 2x$ (or $y = 2x + 0$)

### Point-Slope Form

The other format for straight-line equations is called the "point-slope" form. Suppose that it is wanted to find the equation of a straight line that passes through a known point and has a known slope. For this one, a point $(x_1, y_1)$ and a slope $m$ are given, and they have to be plugged into this formula:

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

Example 2:

1. $y - 4 = -2(x - 1)$

2. $y - 8 = 3(x - 2)$

3. $y - 12 = 4(x - 3)$

Example 3:

Find the equation of a line passing through the point $(4, 2)$ and having a slope of 3.

Solution:

\begin{aligned} \color{red}{x_1} &\color{red}{= 4, \ \ y_1 = 2, \ \ m = 3} \\ y - y_1 &= m \cdot (x - x_1) \\ y - 2 &= 3 (x - 4) \\ y - 2 &= 3x - 12 \\ y &= 3x - 10 \end{aligned}

### Two Point Form

If two points $(x_1, x_2)$ and $(y_1, y_2)$ are available, the two point form equation for a line will be used:

$$y - y_1 = \frac{y_2 - y_1}{x_2 - x_1} \cdot (x - x_1)$$

The slope formula is $m = \frac{y_2 - y_1}{x_2 - x_1}$

Example 4:

Find the equation of the line that passes through the points $(2, 4)$ and $(1, 2)$.

Solution:

\begin{aligned} x_1 &= 3, \ \ y_1 = 4, \ \ x_2 = 1, \ \ y_2 = -5 \\ \color{red}{y - y_1} &\color{red}{= \frac{y_2 - y_1}{x_2 - x_1} \cdot (x - x_1)} \\ y - 4 &= \frac{-5 - 1}{1 - 3}(x - 3) \\ y - 4 &= 3(x - 3) \\ y - 4 &= 3x - 9 \\ y &= 3x - 5 \end{aligned}

Example 5:

Find the equation of a line through the points $(1,2)$ and $(3,1)$. What is its slope? What is its y intercept?

Solution: First find the slope of the line by finding the ratio of the change in y over the change in x. Thus:

$m = \frac{2 - 1}{1 - 3} = - \frac{1}{2}$

Now, the point - slope form can be used to obtain:

\begin{aligned} \color{red}{x_1} &\color{red}{= 1, \ \ y_1 = 2, \ \ m = -\frac{1}{2}} \\ y - y_1 &= m \cdot (x - x_1) \\ y - 2 &= - \frac{1}{2} (x - 1) \\ y - 2 &= - \frac{1}{2} x + \frac{1}{2} \\ y &= - \frac{1}{2} x + \frac{5}{2} \\ y &= - \frac{x}{2} + \frac{5}{2} \end{aligned}

### Standard Form

In the Standard Form of the equation of a straight line, the expression is:

$Ax + By = C$

where $A$ and $B$ are not both equal to zero.

1. $7x + 4y = 6$

2. $2x - 2y = -2$

3. $-4x + 17y = -432$