X-axis

Y-axis

(0,0)

Origin

Quadrant IQuadrant II

Quadrant III Quadrant IV

Rectangular Coordinate System

X-axis

Y-axis

(0,0)

Rectangular Coordinate System

• (1,3)

• (-2,-4)

• (0,5)

• (6, -3)

• (-5,4)

Linear Equations in Two Variable

A linear equation in two variables is an equation of the form Ax + By = C

where A, B, C are real numbers and A & B are both not zero

A linear equation in two variables is an equation of the form Ax + By = C

where A, B, C are real numbers and A & B are both not zero

The graph of any linear equation in two variables is a straight line

Examples of linear equation in two variables :3x + 4y =23X=67z + 4y =16X + 9y = 0

Solutions of Linear Equations in Two Variables

• Example:

2x + y = 3Some Solutions

• Example:

y = 3x - 1Some Solutions

x y0 3

11-12

-1 5

x y

0 -11

52

-1 - 4

2

Graphing Linear Equations in Two Variables

• Graph:

2x + y = 3

31-1

x y

012

-1 5

5

4

3

2

1

-2

-3

-4

-5

y

-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

x

(0, 3)

(1, 1)

(2, -1)

Graphing Linear Equations in Two Variables

• The graph of 2x + y = 3 is a line.

• The solution set of a linear equation in two variables is all the points that lie on the line of its graph

5

4

3

2

1

-2

-3

-4

-5

y

-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

x

X-intercepts and Y-intercepts

• The x-intercept is the point where the line intersects the x-axis

• What is the y-coordinate of the x-intercept?

• The y-intercept is the point where the line intersects the y-axis

• What is the x-coordinate of the y-intercept?

5

4

3

2

1

-2

-3

-4

-5

y

-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

x

2x + y = 3

Y - Interceptsy

x

The y – intercept is the y – value when x = 0.

When using a graph, the y – intercept is where the graph crosses the y – axis.

The y – intercept of the line to the right is …

y – intercept is 4.

To find the y-intercept, let x=0 in the given equation and solve for y. Then (0,y) is the y-intercept.

X - Interceptsy

x

The x – intercept is the x – value when y = 0.

When using a graph, the x – intercept is where the graph crosses the x – axis.

The x – intercept of the line to the right is …

x – intercept is 2.

To find the x-intercept, let y=0 in the given equation and solve for x. Then (x,0) is the x-intercept

Special Cases

• Graph: y = 3

• This is the set of points that have a y-coordinate of 3.

• (x, 3) for all x.

5

4

3

2

1

-2

-3

-4

-5

y

-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

x

The Graph of y= 3 is a horizontal line

Special Cases

• Graph: x = 5

• This is the set of points that have a x-coordinate of 5.

• (5, y) for all y.

5

4

3

2

1

-2

-3

-4

-5

y

-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

x

The graph of x= 5 is a vertical line

Graphing ax + by = c

y

x

What about 3x + 4y = –12 ?

Step 1: Put equation in slope – intercept form

This means solve for y.

334

y x=- -

Step 2: Plot the y – intercept.

Step 3: Starting at the y – intercept, use the slope to find other points.

A slope of means

Down 3, Right 4…or…Up 3, Left 4 (the 3 or 4 can be negative, but not both).

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