Intermediate Algebra/Linear Equations
A linear equation is an equation that forms a line on a graph.
A linear equation in slope-intercept form is one in the form such that is the slope, and is the y-intercept. An example of such an equation is:
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 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:
Which passes through
Substitute and for and , respectively
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:
which passes through the point
Replace and with and , respectively. Simplify. Put into slope-intercept form.
The Standard form of a line is the form of a linear equation in the form of such that and are integers, and .
Converting from slope-intercept form to standard form
Slope-intercept equations can easily be changed to standard form. Consider the equation:
Subtract -3x from each side, satisfying
Multiply the entire equation by , satisfying
and 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