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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.

- 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

- 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.

- 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:

- 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

- 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.

- 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 ┴

Complete the following table:

- 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)