Forms of a Linear Equation
Switch fluently between slope-intercept, point-slope, and standard forms of a linear equation: each form highlights different properties and suits different problem types.
Key Concepts
There are three common forms: Standard Form is $Ax + By = C$. Slope Intercept Form is $y = mx + b$. Point Slope Form is $y y 1 = m(x x 1)$. Each form describes the same line but reveals different information.
Convert $3x + 2y = 12$ to slope intercept form: $2y = 3x + 12$, so $y = \frac{3}{2}x + 6$. Graph $y 3 = \frac{1}{2}(x + 2)$ by plotting the point $( 2, 3)$ and using the slope $m = \frac{1}{2}$ to find another point.
These forms are like different outfits for the same line. Slope Intercept is great for quick graphing, Point Slope is perfect when you know a point and the steepness, and Standard Form keeps everything neat and tidy. You can always switch between outfits by rearranging the equation algebraically.
Common Questions
What are the three main forms of a linear equation and what does each highlight?
Slope-intercept form y=mx+b highlights slope m and y-intercept b directly. Point-slope form y-y1=m(x-x1) is built from a known point and slope. Standard form Ax+By=C with integer coefficients is useful for elimination in systems and finding intercepts easily.
How do you convert between the three linear equation forms?
To go from point-slope to slope-intercept: distribute and simplify to isolate y. From slope-intercept to standard: move the mx term left, multiply to clear fractions, ensure A is positive. From standard to slope-intercept: solve for y by dividing by B.
Which form is best to use when given two points?
Calculate slope from the two points using m=(y2-y1)/(x2-x1), then substitute one point into point-slope form immediately. Convert to slope-intercept or standard form afterward if required. Starting with point-slope is fastest because it uses the given data directly.