The Various Forms of a Linear EquationLearning mathematics is very much like the construct...
The Various Forms of a Linear Equation
Learning mathematics is very much like the construction of a skyscraper. The final height of the skyscraper ultimatel depends on the strength of the foundation and quality of the frame supporting each new floor as it is built. Our previous work with linear functions and their graphs, while having a number of useful applications, is actually the foundation on which much of our future work is built. For this reason, it’s important you gain a certain fluency with linear functions and relationships—even to a point where things come to you effortlessly and automatically. As noted mathematician Henri Lebesque once said, “An idea reaches its maximum level of usefulness only when you understand it so well that it seems like you have always known it. You then become incapable of seeing the idea as anything but a trivial and immediate result.” These formulas and concepts, while simple, have an endless number of significant and substantial applications.
Forms and Formulas
slope formula | point-slope form | slope-intercept form | standard form |
| y − y1 = m(x − x1) | y = mx + b | Ax + By = C |
given any two points on the line | given slope m and any point (x1, y1) | given slope m and y-intercept (0, b) | A, B, and C are integers (used in linear systems) |
Characteristics of Lines
y-intercept | x-intercept | increasing | decreasing |
(0, y) | (x, 0) | m > 0 | m > 0 |
let x = 0, solve for y | let y = 0, solve for x | line slants upward from left to right | line slants downward form left to right |
Relationships between Lines
intersecting | parallel | perpendicular | dependent |
m1 ≠ m2 | m1 = m2, b1 ≠ b2 | m 1m2 = − 1 | m 1 = m2, b1 = b2 |
lines intersect at one point | lines do not intersect | lines intersect at right angles | lines intersect at all points |
Special Lines
Horizontal | vertical | identity |
y = k | x = h | y = x |
horizontal line through k | vertical line through h | the input value identifies the output |
Use the formulas and concepts reviewed here to complete the following exercises.
For the two points given: (a) compute the slope of the line through the points and state whether the line is increasing or decreasing, (b) find the equation of the line in point-slope form, then write the equation in slope-intercept form, and (c) find the x- and y-intercepts and graph the line.
P1(− 2, 5) P2(6,−1)
Step-by-Step Solution
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