A linear equation in two variables is an equation that can be written in the form: $$Ax+By=C$$ where \(A, \: B \) and \( C \) are any real valued constants. This form is called standard form.
Slope is the ratio of how a line is changing vertically to how a line is changing horizontally. Typically, a lower case letter \( m \) is used to denote the slope of a line. In the coordinate plane, if \( \left( x_{i},y_{i} \right) \) are the coordinates of the initial point and \( \left( x_{f},y_{f} \right) \) are the coordinates of the final point, then the slope is calculated by:
$$m=\frac{y_{f}-y_{i}}{x_{f}-x_{i}}$$If Joe fills his 15 gallon car with gas, it costs him $41.85. Suzie paid $22.32 to fill her 8 gallon tank. Determine what the cost per gallon.
The cost per gallon would be the slope of the line connecting the given two points: \( \left( 15, 41.85 \right) \) and \( \left( 8, 22.32 \right) \). Cost is the \( y \)-coordinate and gallons the \( x \)-coordinate. $$m=\frac{41.85-22.32}{15-8} \Rightarrow m=2.79\;\color{green}{\checkmark}$$ The cost per gallon of gas would be $2.79.
The slope-intercept form is the most recognizable forms for lines: \(y=mx+b\) where \(m\) is the slope and the value of \(b\) gives the \(y\)-coordinate of the \(y\)-intercept. However, the \(y\)-intercept is a point in the coordinate plane and should be expressed with coordinates \((0,b)\).
The point-slope form is a less familiar form: \(y-y_{i}=m\left(x-x_{i}\right)\) but has many uses in equation writing. In this form, the values of \(x_{i}\) and \(y_{i}\) are from any point on the line \(\left(x_{i},y_{i}\right)\).
WARNING: Be careful here, the form involves subtraction, and when \(x_{i}\) or \(y_{i}\) is a negative number, it will turn to addition. See the examples below.
Create an equation for a line in slope-intercept form that passes through the points \(\left(-3,8\right)\) and \(\left(6,-2\right)\).
| Step 1: Find the slope: | \(m=\large{\frac{8-(-2)}{-3-6}}\normalsize{\implies m=\large{\frac{10}{-9}}}\) |
| Step 2: Write in point-slope form first: | \(\implies y-8=-\frac{10}{9}\left(x+3\right)\) |
| Step 3: Convert to slope-intercept: | \(\implies y-8=-\frac{10}{9}x-\frac{30}{9}\) OR \(-\frac{10}{3}\) |
| \(\implies y=-\frac{10}{9}x-\frac{10}{3}+8\) OR \(\frac{24}{3}\) | |
| Final answer: | \(\implies y=-\frac{10}{3}x-\frac{14}{3}\;\color{green}{\checkmark}\) |
Note: There is an alternative method to do this problem involving finding \(b\) outright. Ask me about it if you are curious.
In Geometry, two lines are parallel if they do not intersect. In algebra, two lines are parallel if the slopes are EQUAL. Likewise, in Geometry, two lines are perpendicular if the intersection between the lines forms a right angle. In algebra, two lines are perpendicular if the slopes are OPPOSITE RECIPROCALS. The examples that follow show how to create equations for parallel and perpendicular lines that practice the skills from the above example.
For the line \(y=\frac{2}{5}x-12\), create an equation of a line that is parallel to the given line and passes through \(\left(-4,7\right)\).
First, note that the given line is in slope-intercept form, so \(m=\frac{2}{5}\). A line that is parallel will have equal slope. Hence, use \(m=\frac{2}{5}\) and the point-slope form to create an equation: \(y-7=\frac{2}{5}\left(x+4\right)\;\color{green}{\checkmark}\).
Find an equation in standard form for a line perpendicular to \(14x+8y=-12\) that passes through \(\left(3,-10\right)\).
| Rewrite given equation to find the slope: | \(8y=-14x-12\) |
| \(\implies y=-\frac{14}{8}x-\frac{12}{8}\) | |
| \(\implies y=-\frac{7}{4}x-\frac{3}{2}\) so \(m=-\frac{7}{4}\) | |
| Perpendicular means use slope of \(m=\frac{4}{7}\). | |
| Use the point-slope form: | \(y+10=\frac{4}{7}\left(x-3\right)\) |
| Algebra to put into standard form: | \(7\left(y+10\right)=4\left(x-3\right)\) |
| \(\implies7y+70=4x-12\) | |
| \(\implies4x-7y=82\;\color{green}{\checkmark}\) | |