Intermediate Algebra/Linear Equations

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Linear Equations[edit]

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

Slope-Intercept form[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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