3 2 linear equations and lines

Linear Equations and Lines
Frank Ma © 2011

In the rectangular coordinate system, ordered pairs (x, y)’s correspond to locations of points.

In the rectangular coordinate system, ordered pairs (x, y)’s correspond to locations of points. Collections of points may be specified by the mathematics relations between the
x-coordinate and the y coordinate.

x-coordinate and the y coordinate. The plot of points that fit a given relation is called the graph of that relation.

x-coordinate and the y coordinate. The plot of points that fit a given relation is called the graph of that relation. To make a graph of a given mathematics relation, make a table of points that fit the description and plot them.

Example A. Graph the points (x, y) where x = –4

(y can be anything).

Make a table of ordered pairs of points that fit the description
x = –4.

x = –4.

x = –4.
Graph of x = –4

Example B. Graph the points (x, y) where y = x.

Make a table of points that fit the description y = x.

Make a table of points that fit the description y = x. To find one such point, we set one of the coordinates to be a number, any number, than use the relation to find the other coordinate.

Make a table of points that fit the description y = x. To find one such point, we set one of the coordinates to be a number, any number, than use the relation to find the other coordinate. Repeat this a few times.

Graph the points (x, y) where y = x

First degree equation in the variables x and y are equations that may be put into the form Ax + By = C where A, B, C are numbers.

First degree equation in the variables x and y are equations that may be put into the form Ax + By = C where A, B, C are numbers. First degree equations are the same as linear equations.

First degree equation in the variables x and y are equations that may be put into the form Ax + By = C where A, B, C are numbers. First degree equations are the same as linear equations. They are called linear because their graphs are straight lines.

First degree equation in the variables x and y are equations that may be put into the form Ax + By = C where A, B, C are numbers. First degree equations are the same as linear equations. They are called linear because their graphs are straight lines. To graph a linear equation, find a few ordered pairs that fit the equation.

First degree equation in the variables x and y are equations that may be put into the form Ax + By = C where A, B, C are numbers. First degree equations are the same as linear equations. They are called linear because their graphs are straight lines. To graph a linear equation, find a few ordered pairs that fit the equation. To find one such ordered pair, assign a value to x,

First degree equation in the variables x and y are equations that may be put into the form Ax + By = C where A, B, C are numbers. First degree equations are the same as linear equations. They are called linear because their graphs are straight lines. To graph a linear equation, find a few ordered pairs that fit the equation. To find one such ordered pair, assign a value to x, plug it into the equation and solve for the y

First degree equation in the variables x and y are equations that may be put into the form Ax + By = C where A, B, C are numbers. First degree equations are the same as linear equations. They are called linear because their graphs are straight lines. To graph a linear equation, find a few ordered pairs that fit the equation. To find one such ordered pair, assign a value to x, plug it into the equation and solve for the y (or assign a value to y and solve for the x).

First degree equation in the variables x and y are equations that may be put into the form Ax + By = C where A, B, C are numbers. First degree equations are the same as linear equations. They are called linear because their graphs are straight lines. To graph a linear equation, find a few ordered pairs that fit the equation. To find one such ordered pair, assign a value to x, plug it into the equation and solve for the y (or assign a value to y and solve for the x). For graphing lines, find at least two ordered pairs.

Example C.
Graph the following linear equations.
y = 2x – 5

Example C.
y = 2x – 5
Make a table by selecting a few numbers for x.

Example C.
y = 2x – 5
Make a table by selecting a few numbers for x. For easy caluation we set x = -1, 0, 1, and 2.

Example C.
y = 2x – 5
Make a table by selecting a few numbers for x. For easy caluation we set x = -1, 0, 1, and 2. Plug each of these value into x and find its corresponding y to form an ordered pair.

For y = 2x – 5:

For y = 2x – 5:
If x = -1, then
y = 2(-1) – 5

For y = 2x – 5:
If x = -1, then
y = 2(-1) – 5 = -7

For y = 2x – 5:
If x = -1, then
y = 2(-1) – 5 = -7
If x = 0, then
y = 2(0) – 5

For y = 2x – 5:
If x = -1, then
y = 2(-1) – 5 = -7
If x = 0, then
y = 2(0) – 5 = -5

For y = 2x – 5:
If x = -1, then
y = 2(-1) – 5 = -7
If x = 0, then
y = 2(0) – 5 = -5
If x = 1, then
y = 2(1) – 5 = -3
If x = 2, then
y = 2(2) – 5 = -1

b. -3y = 12

b. -3y = 12
Simplify as y = -4
Make a table by
selecting a few numbers for x.

b. -3y = 12
Simplify as y = -4
Make a table by
However, y = -4 is always.

b. -3y = 12
Simplify as y = -4
Make a table by
c. 2x = 12

b. -3y = 12
Simplify as y = -4
Make a table by
c. 2x = 12
Simplify as x = 6.

b. -3y = 12
Simplify as y = -4
Make a table by
c. 2x = 12
Simplify as x = 6
Make a table.
However the only selction for x is x = 6

b. -3y = 12
Simplify as y = -4
Make a table by
c. 2x = 12
Simplify as x = 6
Make a table.
However the only selction for x is x = 6 and y could be any number.

Summary of the graphs of linear equations:

a. y = 2x – 5

a. y = 2x – 5
If both variables
x and y are
present in the
equation, the graph is a
tilted line.

a. y = 2x – 5 b. -3y = 12
If both variables
x and y are
present in the
tilted line.

a. y = 2x – 5 b. -3y = 12
If the equation has only y (no x), the graph is a horizontal line.
If both variables
x and y are
present in the
tilted line.

a. y = 2x – 5 b. -3y = 12 c. 2x = 12
If both variables
x and y are
present in the
tilted line.

a. y = 2x – 5 b. -3y = 12 c. 2x = 12
If the equation has only x (no y), the graph is a
vertical line.
If both variables
x and y are
present in the
tilted line.

The x-Intercepts is where the line crosses the x-axis;

The x-Intercepts is where the line crosses the x-axis. We set y = 0 in the equation to find the x-intercept.

The y-Intercepts is where the line crosses the y-axis;

The y-Intercepts is where the line crosses the y-axis. We set x = 0 in the equation to find the y-intercept.

Since two points determine a line, an easy method to graph linear equations is the intercept method,

Since two points determine a line, an easy method to graph linear equations is the intercept method, i.e. plot the x-intercept and the y intercept and the graph is the line that passes through them.

Example C. Graph 2x – 3y = 12
by the intercept method.

y-int
x-int

If x = 0, we get
2(0) – 3y = 12
y-int
x-int

If x = 0, we get
2(0) – 3y = 12
so y = -4
y-int
x-int

If x = 0, we get
2(0) – 3y = 12
so y = -4
y-int
If y = 0, we get
2x – 3(0) = 12
x-int

If x = 0, we get
2(0) – 3y = 12
so y = -4
y-int
If y = 0, we get
2x – 3(0) = 12
so x = 6
x-int

3 2 linear equations and lines

by math123a on Jun 22, 2011

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3 2 linear equations and lines — Presentation Transcript