# Intermediate Algebra/Linear Equations

## Linear Equations

A linear equation is an equation that forms a line on a graph.

### Slope-Intercept form

A linear equation in slope-intercept form is one in the form $y = mx + b$ such that $m$ is the slope, and $b$ is the y-intercept. An example of such an equation is:
$y = 3x - 1$

#### Finding y-intercept, given slope and a point

The y-intercept of an equation is the point at which the line produced touches the y-axis, or the point where $x = 0$ This can be very useful. If we know the slope, and a point which the line passes through, we can find the y-intercept. Consider:

$y = 3x + b$ Which passes through $(1,2)$
$2 = 3(1) + b$ Substitute $2$ and $1$ for $x$ and $y$, respectively
$2 = 3 + b$ Simplify.
$-1 = b$
$y = 3x - 1$ Put into slope-intercept form.

#### Finding slope, given y-intercept and a point

The slope of a line is defined as the amount of change in x and y between two points on the line.

If we know the y-intercept of the line, and a point on the line, we can easily find the slope. Consider:

$y = mx + 4$ which passes through the point $(2,1)$
$y = mx + 4$
$1 = 2m + 4$ Replace $x$ and $y$ with $1$ and $2$, respectively. $-3 = 2m$ Simplify. $-3/2 = m$$y = -3/2x + 4$ Put into slope-intercept form.

### Standard form

The Standard form of a line is the form of a linear equation in the form of $Ax + By = C$ such that $A$ and $B$ are integers, and $A > 0$.

#### Converting from slope-intercept form to standard form

Slope-intercept equations can easily be changed to standard form. Consider the equation:
$y = 3x - 1$
$-3x + y = -1$ Subtract -3x from each side, satisfying $Ax + By = C$
$3x - y = 1$ Multiply the entire equation by $-1$, satisfying $A > 0$
$A$ and $B$ are already integers, so we don't have to worry about changing them.

#### Finding the slope of an equation in standard form

In the standard form of an equation, the slope is always equal to $\frac {-A}{B}$

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