Cartesian Plane and Linear Equations in Two VariablesMath 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:

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.

• 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)

Complete the table of values for each equation:

• y = 2x – 10 x + 3y = 9

x y

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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.

Graph: 2x + y = 4 10

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Graph: y= 3x-1 10

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Graph: y= 2x 10

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Graph: 15= -5y + 3x 10

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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.

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.

Graph: x = 4 10

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Graph: y= -2 10

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Graph: 3x = -15 10

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Graph: y + 3 = 4 10

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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)

• Find the slopes of the lines below:

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= -3 3y -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.

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

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 ┴

Slope // Slope ┴ slope

a.

b.

5

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0

Complete the following table:

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2

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)