Cartesian plane and linear equations in two variables
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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.

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Cartesian Plane and Linear Equations in Two Variables

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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)


Complete the table of values for each equation:

  • 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.


Graph: 2x + y = 4


Graph: y= 3x-1


Graph: y= 2x


Graph: 15= -5y + 3x


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


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


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


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


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


Graph: y= -2


Graph: 3x = -15


Graph: y + 3 = 4


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= -33y -2 = 4


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 ┴


Complete the following table:


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

  • a. 2y = 4x + 7b. 5x – 10y = 6

    y – 2x = -3y = 2x + 7

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

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


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