Lesson Objectives

- Demonstrate an understanding of Linear Equations in Two Variables
- Demonstrate an understanding of an Ordered Pair (x,y)
- Learn how to create a Cartesian Coordinate Plane
- Learn how to identify the Quadrants in the Cartesian Coordinate Plane
- Learn how to plot an ordered pair (x,y) on the Cartesian Coordinate Plane

## How to Plot an Ordered Pair

In the last lesson, we introduced a new type of equation known as a linear equation in two variables. As an example, suppose we came across the equation:

5x - 2y = 20

For this type of equation, we see that we have two variables, x, and y. When we work with a linear equation in two variables, we have an infinite number of solutions. These solutions are most commonly written as (x,y) values known as an ordered pair. For our above equation, we can show that:

(0,-10), (4,0), and (2,-5) are solutions. Just like with a linear equation in one variable, we can plug in for the variables and simplify. We know our solution is correct when each side simplifies to the same value. Let's take a look at our three ordered pairs as solutions to our equation.

(0,-10)

Plug in a 0 for x and a (-10) for y:

5(0) - 2(-10) = 20

20 = 20

(0,-10) is a solution to our equation.

(4,0)

Plug in a 4 for x and a 0 for y:

5(4) - 2(0) = 20

20 = 20

(4,0) is a solution to our equation.

(2,-5)

Plug in a 2 for x and a (-5) for y:

5(2) - 2(-5) = 20

10 + 10 = 20

20 = 20

(2,-5) is a solution to our equation.

I: (+,+) » x and y values are positive

II: (-,+) » x values are negative and y values are positive

III: (-,-) » x and y values are negative

IV: (+,-) » x values are positive and y values are negative

Example 1: Plot each ordered pair and determine its quadrant.

(2,5)

To plot the point (2,5), we want an x location of 2. This means we move 2 units to the right on the x-axis (horizontal axis). We want a y location of 5, this means we move 5 units up on the y-axis (vertical axis). We will place and label a point at that location. Additionally, both x and y values are positive. This means our point lies in quadrant I.

Example 2: Plot each ordered pair and determine its quadrant.

(-4,-7)

To plot the point (-4,-7), we want an x location of -4. This means we move 4 units left on the x-axis (horizontal axis). We want a y location of -7, this means we move 7 units down on the y-axis (vertical axis). We will place and label a point at that location. Additionally, both x and y values are negative. This means our point lies in quadrant III.

Example 3: Plot each ordered pair and determine its quadrant.

(7,-3)

To plot the point (7,-3), we want an x location of 7. This means we move 7 units right on the x-axis (horizontal axis). We want a y location of -3, this means we move 3 units down on the y-axis (vertical axis). We will place and label a point at that location. Additionally, the x value is positive, while the y value is negative. This means our point lies in quadrant IV.

5x - 2y = 20

For this type of equation, we see that we have two variables, x, and y. When we work with a linear equation in two variables, we have an infinite number of solutions. These solutions are most commonly written as (x,y) values known as an ordered pair. For our above equation, we can show that:

(0,-10), (4,0), and (2,-5) are solutions. Just like with a linear equation in one variable, we can plug in for the variables and simplify. We know our solution is correct when each side simplifies to the same value. Let's take a look at our three ordered pairs as solutions to our equation.

(0,-10)

Plug in a 0 for x and a (-10) for y:

5(0) - 2(-10) = 20

20 = 20

(0,-10) is a solution to our equation.

(4,0)

Plug in a 4 for x and a 0 for y:

5(4) - 2(0) = 20

20 = 20

(4,0) is a solution to our equation.

(2,-5)

Plug in a 2 for x and a (-5) for y:

5(2) - 2(-5) = 20

10 + 10 = 20

20 = 20

(2,-5) is a solution to our equation.

### Cartesian Coordinate Plane

In the next section, we will start to graph linear equations in two variables. In order to perform this action, we must first understand how the coordinate plane works. The Cartesian Coordinate Plane, also known as the Rectangular Coordinate Plane is made up of two number lines, one vertical and one horizontal: The horizontal axis is known as the "x" axis. We can see the blue "x" label at the far right of the horizontal number line. We should be fairly familiar with this horizontal number line. We have used this type of number line throughout pre-algebra and algebra to show integers, add integers, graph inequalities,...etc. We know that numbers increase as we move to the right and numbers decrease as we move to the left. New to us is the vertical number line, which is known as the "y" axis. We can see the purple "y" label at the top of the vertical number line. Numbers on the vertical number line increase as we move up and decrease as we move down. The origin is the point where both number lines intersect. This spot has an ordered pair of (0,0), which means x = 0, and y = 0. The origin is the exact center of our coordinate plane: The coordinate plane is split up into four quadrants. These quadrants are labeled with Roman Numerals as: I, II, III, and IV. Quadrant I starts at the top right and the rest are arranged moving counter-clockwise. It's important to note the values in each quadrant:I: (+,+) » x and y values are positive

II: (-,+) » x values are negative and y values are positive

III: (-,-) » x and y values are negative

IV: (+,-) » x values are positive and y values are negative

### How to Plot an Ordered Pair

An ordered pair is often referred to as a point. When we plot a point (ordered pair), we are just finding the meeting point of the x location and the y location. We will draw a closed circle or dot at that location. In many cases, we will label the point with its ordered pair (x,y). Let's look at a few examples.Example 1: Plot each ordered pair and determine its quadrant.

(2,5)

To plot the point (2,5), we want an x location of 2. This means we move 2 units to the right on the x-axis (horizontal axis). We want a y location of 5, this means we move 5 units up on the y-axis (vertical axis). We will place and label a point at that location. Additionally, both x and y values are positive. This means our point lies in quadrant I.

Example 2: Plot each ordered pair and determine its quadrant.

(-4,-7)

To plot the point (-4,-7), we want an x location of -4. This means we move 4 units left on the x-axis (horizontal axis). We want a y location of -7, this means we move 7 units down on the y-axis (vertical axis). We will place and label a point at that location. Additionally, both x and y values are negative. This means our point lies in quadrant III.

Example 3: Plot each ordered pair and determine its quadrant.

(7,-3)

To plot the point (7,-3), we want an x location of 7. This means we move 7 units right on the x-axis (horizontal axis). We want a y location of -3, this means we move 3 units down on the y-axis (vertical axis). We will place and label a point at that location. Additionally, the x value is positive, while the y value is negative. This means our point lies in quadrant IV.

#### Skills Check:

Example #1

Determine the quadrant. $$(-5, 7)$$

Please choose the best answer.

A

Quadrant I

B

Quadrant II

C

Quadrant III

D

Quadrant IV

Example #2

Determine the ordered pair of the given point.

Please choose the best answer.

A

$$(-8,6)$$

B

$$(6,-8)$$

C

$$(-6,-8)$$

D

$$(-8,-6)$$

E

$$[-8, 0), (0,-6]$$

Example #3

Determine the ordered pair of the given point.

Please choose the best answer.

A

$$(0,-5],[-9,0)$$

B

$$(-9,-5)$$

C

$$(9,5)$$

D

$$(-5,-9)$$

E

$$(5,9)$$

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