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**Cartesian Plane and Linear Equations in Two Variables**

Math 021

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**The Cartesian Plane (coordinate grid) is a graph used to show a relationship between two variables.**

The horizontal axis is called the x-axis. The vertical axis is called the y-axis. The point of intersection of the x-axis and y-axis is called the origin. The axes divide the Cartesian Plane into four quadrants. An ordered pair is a single point on the Cartesian Plane. Ordered pairs are of the form (x,y) where the first value is called the x-coordinate and the second value is called the y-coordinate.

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Examples – Plot each if the following ordered pairs on the Cartesian Plane and name the quadrant it lies in: a. A = (3, 4) b. B = (-2, 1) c. C = (7, -3) d. D = (-4, -2) e. E = (0, 5) f. F = (-1, 0)

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**Linear Equations in Two Variables**

A linear equation in two variables is an equation of the form Ax + By = C where A, B, and C are real numbers. The form Ax + By = C is called the standard form of a linear equation in two variables. An ordered pair is a solution to a linear equation in two variables if it satisfies the equation when the values of x and y are substituted.

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**Examples – Determine if the ordered pair is a solution to each linear equation:**

a. 2x – 3y = 6; (6, 2) b. y = 2x + 1; (-3, 5) c. 2x = 2y – 4; (-2, -8) d. 10 = 5x + 2y; (-4, 15)

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**Examples – Find the missing coordinate in each ordered par given the equation:**

a. -7y = 14x; (2, __ ) b. y = -6x + 1; ( ____, -11) c. 4x + 2y = 8; (1, __ ) d. x – 5y = -1; ( ____, -2)

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**Complete the table of values for each equation:**

y = 2x – 10 x + 3y = 9 x y 4 -20 5 x y 6 4

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**Graphing Linear Equations in Two Variables**

The graph of an equation in two variables is the set of all points that satisfies the equation. A linear equation in two variables forms a straight line when graphed on the Cartesian Plane. A table of values can be used to generate a set of coordinates that lie on the line.

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Graph: 2x + y = 4

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Graph: y= 3x-1

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Graph: y= 2x

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Graph: 15= -5y + 3x

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**Intercepts An intercept is a point on a graph which crosses an axis.**

An x-intercept crosses the x-axis. The y-coordinate of any x-intercept is 0. A y-intercept crosses the y-axis. The x-coordinate of any y-intercept is 0.

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**Graph by Finding Intercepts: 3x – 2y = 12**

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**Graph by Finding Intercepts: y= -2x + y**

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**Graph by Finding Intercepts: 4x + 3y = -12**

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**Graph by Finding Intercepts: 3x – 5y = -15**

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**Horizontal and Vertical Lines**

A horizontal line is a line of the form y = c, where c is a real number. A vertical line is a line of the form x = c, where c is a real number.

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Graph: x = 4

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Graph: y= -2

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Graph: 3x = -15

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Graph: y + 3 = 4

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Slope of a Line The slope of a line is the degree of slant or tilt a line has. The letter “m” is used to represent the slope of a line. Slope can be defined in several ways: Examples - Find the slope of each line: a. Containing the points (3, -10) and (5, 6) b. Containing the points (-4, 20) and (-8, 8)

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**Find the slopes of the lines below:**

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**Slopes of Horizontal & Vertical Lines **

The slope of any horizontal line is 0 The slope of any vertical line is undefined Examples – Graph each of the following lines then find the slope x= y -2 = 4

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**Slope-Intercept form of a Line**

The slope-intercept form of a line is y = mx + b where m is the slope and the coordinate (0,b) is the y-intercept. The advantage equation of a line written in this form is that the slope and y-intercept can be easily identified.

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**Examples – Find the slope and y-intercept of each equation:**

a. y = 3x – 2 b. 4y = 5x + 8 c. 4x + 2y = 7 d. 5x – 7y = 11

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**Parallel and Perpendicular Slopes**

Two lines that are parallel to one another have the following properties They will never intersect They have the same slopes They have different y-intercepts Parallel lines are denoted by the symbol // Two lines that are perpendicular to one another have the following properties: They intersect at a angle The have opposite and reciprocal slopes Perpendicular lines are denoted by the symbol ┴

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**Complete the following table:**

Slope // Slope ┴ slope a. b. 5 c.

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**Examples – Determine if each pair of lines is parallel, perpendicular, or neither:**

a. 2y = 4x + 7 b. 5x – 10y = 6 y – 2x = y = 2x + 7 c. 3x + 4y = 3 d. Line 1 contains points (3,1) and (2,7) 4x + 5y = Line 2 contains points (8,5) and (2,4)

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