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# Cartesian Plane and Linear Equations in Two Variables - PowerPoint PPT Presentation

Cartesian Plane and Linear Equations in Two Variables. Math 021. 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.

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### Cartesian Plane and Linear Equations in Two Variables

Math 021

• 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.

Examples – Plot each if the following ordered pairs on the Cartesian Plane and name the quadrant it lies in:

• 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.

• Anordered pairis a solution to a linear equation in two variables if it satisfies the equation when the values of x and y are substituted.

• 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)

• 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)

• y = 2x – 10x + 3y = 9

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 straightline when graphed on the Cartesian Plane.

• A table of values can be used to generate a set of coordinates that lie on the line.

• 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.

• 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.

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)

• 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= -3 3y -2 = 4

• 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.

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

• 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 ┴

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

• a. 2y = 4x + 7 b. 5x – 10y = 6

y – 2x = -3 y = 2x + 7

• c. 3x + 4y = 3 d. Line 1 contains points (3,1) and (2,7)

4x + 5y = -1 Line 2 contains points (8,5) and (2,4)