Math 31 - Ch 3. - Graphing Linear Equations in Two Variables

Using Intercepts to Graph Ax + By = C

1. Find the x-intercept. Let y = 0 and solve for x. 2. Find the y-intercept. Let x = 0 and solve for y. 3. Find a checkpoint, a third ordered-pair solution. 4. Graph the equation by drawing a line through the three points.

Slope and Parallel Lines

1. If two nonvertical lines are parallel, then they have the same slope. 2. If two distinct nonvertical lines have the same slope, then they are parallel. 3. Two distinct vertical lines, each with undefined slope, are parallel.

Slope and Perpendicular Lines

1. If two nonvertical lines are perpendicular, then the product of their slopes is -1. 2. If the product of the slopes of two lines is -1, then the lines are perpendicular. 3. A horizontal line having zero slope is perpendicular to a vertical line having undefined slope.

How to Graph an Inequality in Two Variables

1. Replace the inequality symbol with an equal sign .... 2. Graph the corresponding linear requisition. (Using method for Standard Form.)) 3.1 Draw a solid line ____ IF original inequality contains a </ or \> ect symbol. 3.2 Draw a dashed line - - - IF the original inequality contains a < or > symbol. 4.1 Choose a test point from one of the half planes. (Do not choose a point IN the line) 4.2 Sub the coordinates of the test point into the inequality 5.1 If TRUE statement results - shade half the plane WITH the test point. 5.2 If false statement, shade half plane without test point. NOTE - Test points are only necessary with *Standard Form*.

Graphing Linear Inequalities Without Test Points

If using y>mx+b or y<mx+b you don't need a test point. - If y>mx+b = Shade above the line - If y<mx+b = Shade below the line

Plotting/Plot

Locating points corresponding with Ordered Pairs.

Lines M in Linear Equations

M determines the line's steepness. Any real numbers can be used for M.

Ax + By = 0

Passes through the origin. Can be graphed using the origin as one point on the line. - Find two other points by finding two other solutions of the equation. - Select values for either variable, other than 0, and find corresponding values for the other variable.

Graph of the Equation

Set of all points whose coordinates satisfy the equation.

Slope

The steepness of a line on a graph

Rise

The vertical change between any two points on a line

X-Intercept

The x-coordinate of a point where the graph intersects the x-axis. The y-coordinate corresponding to an x-intercept is *always zero.*

m Variable

This is the slope of the line represented.

Finding Ordered Pairs that are Solutions of an Equation in Two Variables

- Select a value for one of the variables - Substitute that value into the equation and find the corresponding value of the other variable - Use the values of the two variables to form an ordered pair (x, y). This pair of a solution of the equation.

Graphing Ax + By = C by Using the Slope and y-Intercept

1. Begin by solving Ax + By = C for y. This will put the equation in slope-intercept form. 2. Then use the three-step procedure to graph the equation.

Point Plotting Method

1. Find several ordered pairs that are solutions of the equation. 2. Plot these ordered pairs as points in the Rectangular Coordinate System. 3. Connect the Points with a smooth curve out line.

Intercepts of a line

A point where a line intersects a coordinate axis is called an intercept.

Y- Intercept

The y-coordinate of a point where a graph crosses the y-axis. *The x-coordinate corresponding to a y-intercept is always zero.*

Graphing Horizontal or Vertical Inequalities

These do not require test points, either. Vertical Lines: a = x - If x>a - shade to right of x=a - If x<a - shade to the left of x=a Horizontal Lines: b = y - If y>b shade above the line - If y<b shade below line

Parallel

Two non intersecting lines that lie in the same plane. The have the same slope. The same "steepness."

y-axis

Vertical axis on a coordinate plane. Positive numbers above. Negative numbers below.

Point-Slope Form of the Equation of a (nonvertical) Line

use only if: 1. we know slope 2. and point not containing y-intercept y - y1 = m(x - x1) you will never substitute numbers for x and y. only for x1 and y1 and m.

Slope-Intercept Form of the Equation of a (non vertical) Line

y = mx + b use with slope and y intercept

Standard Form

Ax+By=C

Ordered Pair of Real Numbers

Each point in the Rectangular Coordinate System corresponds to an ordered Pair. (0, 0)

Linear Equations in Two Variables

Equations like y = 3x and y = 2x

X-Coordinate

First number of the Ordered Pair. Horizontal locations.

Quadrants of the Rectangular Coordinate System

Four regions of the Rectangular Coordinate System which divide into I, II, III, IV.

x-axis

Horizontal number line in a coordinate plane. Positive numbers to the right. Negative numbers to the left.

Ax + By = C

All equations in the form of Ax + By = C are straight lines when graphed as long as A and B are not both zero.

Solution of an Inequality in Two Variables

An ordered pair or ordered pairs that make the inequality true.p

Negative Reciprocals

Reciprocals with opposite signs. Two nonvertical line are perpendicular if the slope of one is the negative reciprocal of the slope of the other.

Scatter Plot

Set of points representing data.

Graph of Inequality in Two Variables

Similar to graphing an equation in two variables.

Calculating Slope

Slope = Rise/Run

Run

The horizontal change between any two points on a line

Origin

The point of intersection of the axes. X and y axes zero points.

Y-Coordinate

The second number in an Ordered Pair. Vertical locations.

Linear Inequality in Two Variables

ax + by > c; ax + by < c; y > mx + b; y < mx + b

Lines b in Linear Equations

b determines the pint where the line crosses he y-axis. Any real numbers can be used for b.

Equations of Horizontal and Vertical Lines

If A or B in the equation Ax + By = C turns out to be zero you end up with vertical lines.

Graphing y = mx + b by Using the Slope and y-Intercept

If a line's equation is written with y isolated on one side, we can use the y-intercept and the slope to obtain its graph. 1. Plot the point containing the y-intercept on the y-axis. This is the point (0,b). 2. Obtain a second point using the slope, m. Write m as a fraction, and use the rise over run, starting at the point on the y-axis, to plot this point. 3. Draw a line through the two points. Draw arrowheads at the ends of the line to show that the line continues indefinitely in both directions.

Perpendicular

Two lines that intersect at a right angle, 90 degrees.

Checkpoint

When graphing using intercepts it's always a good idea to use a third point *checkpoint* before drawing a line. 1. Select a value for either variable, other than 0. 2. Find the corresponding value for the other variable. Checkpoint should lie on the same line as the x and y intercepts. Recheck your work if it does not.

A Solution of an Equation in Two Variables (x and y)

When the x-coordinate and y-coordinate are substituted for x and y in the equation, we get a true statement.