Presentation on theme: "Cartesian Plane and Linear Equations in Two Variables"— Presentation transcript:
1 Cartesian Plane and Linear Equations in Two Variables Math 021
2 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.
3 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)
4 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.
5 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)
6 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)
7 Complete the table of values for each equation: y = 2x – 10 x + 3y = 9xy4-205xy64
8 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.
13 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.
23 Slope of a LineThe 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)
25 Slopes of Horizontal & Vertical Lines The slope of any horizontal line is 0The slope of any vertical line is undefinedExamples – Graph each of the following lines then find the slopex= y -2 = 4
26 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.
27 Examples – Find the slope and y-intercept of each equation: a. y = 3x – 2b. 4y = 5x + 8c. 4x + 2y = 7d. 5x – 7y = 11
28 Parallel and Perpendicular Slopes Two lines that are parallel to one another have the following propertiesThey will never intersectThey have the same slopesThey have different y-interceptsParallel lines are denoted by the symbol //Two lines that are perpendicular to one another have the following properties:They intersect at a angleThe have opposite and reciprocal slopesPerpendicular lines are denoted by the symbol ┴
29 Complete the following table: Slope// Slope┴ slopea.b.5c.
30 Examples – Determine if each pair of lines is parallel, perpendicular, or neither: a. 2y = 4x + 7 b. 5x – 10y = 6y – 2x = y = 2x + 7c. 3x + 4y = 3 d. Line 1 contains points (3,1) and (2,7)4x + 5y = Line 2 contains points (8,5) and (2,4)