Summary of Linear Equations

[Pages:2]LINEAR EQUATIONS OF TWO VARIABLES

GENERAL FORM: Ax + By = C Solving the Equation = Graphing a Line

Graphing occurs in the Cartesian plane: Two ordered pairs, plotted as points, are sufficient to determine a line.

Methods of Solving:

Lay It On the Table!

You could pick any values for x...

x y 0 1 2

the origin

y - axis

Intercept Method: Fill in (0, _ ) and ( _ ,0) and connect those points.

x - axis

Slope-Intercept Formula: Solve for y; isolate as y = mx + b

Then start at (0, b) and follow the slope m as "rise over run."

Slope = m y 2 y1 y x2 x1 x

"rise"

"run"

The slope formula, m y2 y1 , is the source for an alternative: x2 x1

If x1, y1 is a fixed point on a line and x, y is any other point, then:

m y y1 x x1

m x x1 y y1 This is called the Point-Slope Formula.

 Creating the Equation = Modeling a Line

In order to derive the equation of a line, one must have two necessary parameters: the slope and a point.

In some sense this makes Point-Slope Form the ideal tool to derive an equation, but it isn't the only tool.

Possible Cases:

Given Slope and a Point:

Use the Slope-Intercept Formula:

y = mx +b OR

1. Substitute slope for m.

2. Substitute the point as x and y ; solve for b

Use the Point-Slope Formula:

mx x1 y y1

1. Substitute slope for m

2. Substitute point as x1, y1

Given Two Points:

Compute the slope using m y2 y1 Then, proceed as above... x2 x1

Use the Slope-Intercept Formula:

y = mx +b OR

1. Substitute slope for m.

2. Substitute a point as x and y ; solve for b

Use the Point-Slope Formula:

mx x1 y y1

1. Substitute slope for m

2. Substitute a point as x1, y1

Given Paired Data (applications/word problems):

Paired data should form "natural" ordered pairs.

Often two pairs of data are given, although it isn't uncommon for one pair to be provided along with a fixed rate ? which should be designated as the slope. If two data pairs are given, find the rate/slope first.

Then, proceed...as above! Good luck!

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