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Analytic Geometry: (lesson 2 of 4)

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).

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 $y = 2x + 0$)

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:

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

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}

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