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## Graphing Linear Equations

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**X and Y Coordinates**• A coordinate or ordered-pair notation is always written as (x-coordinate, y-coordinate) or (x, y). • They are always in parentheses with the value of x always first and the value of y always second. That will never change. • Coordinates allow us to identify a specific place on a graphing system called the rectangular coordinate system. • A linear equation in two variables looks like:2x + y = 9 because it contains two variables, namely x and y.**Solutions of Equations in Two Variables**• Determine if the following ordered pairs are a solution of the equation: x – 2y = 6 • (6, 0) (0, 3) (1, -5/2) Substitute the ordered pairs into the given equation: 6 – 2(0) = 6 0 – 2(3) = 6 1 – 2(-5/2) = 6 6 – 0 = 6 -6 = 6 1 + 5 = 6 6 = 6 -6 = 6 6 = 6 yes no yes**You try it….**• Determine which of the ordered pairs are solutions for the given equation:2x – 3y = 6 (0, 2) (3, 0), (6, 2) (0, -2) • Do the work first and then click the mouse button to see if you got them right!**Did you get them right?**2(0) – 3(2) = 6 2 – 6 = 6 -4 = 6 No 2(3) – 3(0) = 6 6 – 0 = 6 6 = 6 Yes 2(6) – 3(2) = 6 12 – 6 = 6 6 = 6 Yes 2(0) – 3(-2) = 6 0 + 6 = 6 6 = 6 Yes What this tells us is that the coordinates (3, 0), (6, 2), and (0, -2) can all be plotted on a graphing system and we can connect them together in a straight line. The point (0, 2) will not be on the line.**Completing Ordered Pairs**• When finding a missing coordinate in a pair, you substitute the known coordinate into the equation and solve the equation to find the missing coordinate. For instance: x + y = 12 (4, ), ( , 5), (0, ), ( , 0) 4 + y = 12 y = 8 x + 5 = 12 x = 7 0 + y = 12 y = 12 x + 0 = 12 x = 12 The complete coordinates are then: (4, 8), (7, 5), (0, 12), and (12, 0)**You try it …..**• 5x – y = 15 ( ,0), (2, ), (4, ), ( , -5) 5(2) – y = 15 10 – y = 15 -y = 5 y = -5 5(4) – y = 15 20 – y = 15 -y = -5 y = 5 5x – 0 = 15 5x = 15 x = 3 5x – (-5) = 15 5x + 5 = 15 5x = 10 x = 2 The complete ordered pairs would be (3, 0), (2, -5), (4, 5), (2, -5). How did you do? Keep practicing if you are having problems.**The Rectangular Coordinate System**Graphing**Rectangular Coordinate System**The point (1, 5) is in line with the 1 of the x-axis and 5 on the y-axis. Quadrant 2 (-, +) Quadrant 1 (+, +) • (1,5) • x-axis Where the x-axis and y-axis meet is called the origin. Quadrant 3 (-, -) Quadrant 4 (+, -) y-axis**Completing a Table of Values**• Complete the table for the equation 2x + y = 4 Substitute the known value from the table into the equation and solve for the other variable. 2(0) + y = 4 2x + 0 = 4 0 + y = 4 2x = 4 y = 4 x = 2 What is the value of x when y = 2? Click here to see if you were right.**Graphing Linear Equations**• A linear equation in two variables is an equation that can be written in the form ax + by = c (a, b, and c are coefficients) • Graph the linear equation x + y = 7Create a Table of Values and choose numbers for x or y. In this example we will substitute values for x. You can substitute any value you want and can choose either the x or y value to assign numbers to, it’s your choice. Hint: It’s easiest to choose the numbers 0, 1, or 2 when filling in the table of values. If fractions are involved, then use multiplies of the denominator.**Graphing Linear Equations (cont’d)**• Complete a table of values for the equation x + y = 70 + y = 7 1 + y = 7 2 + y = 7y = 7 y = 6 y = 5 The completed table of values will give us the coordinates that we will use to plot the line on the rectangular coordinate system. (0, 7) (1, 6) (2, 5)**Graphing Linear Equations (cont’d)**After you have found the points from the Table of Values and you plot them on the graph, your points must line up in a straight line. If they do not, then you made a mistake in your math and need to go back and check to see what went wrong. (0, 7) (1, 6) (2, 5) •(0, 7) •(1, 6) •(2, 5)**Vertical and Horizontal Lines**• Vertical lines have an equation of x=c. • Horizontal lines have an equation of y=c. y = c x = c**Examples of Horizontal and Vertical Lines**The red line means that whenever y is equal to any number, the x value will always be 5. Examples of coordinates would be: (5, 7), (5, 10), (5,-14), (5, -1) The value of x is always 5 which means it is a vertical line. • x = 5Since x is equal to a coefficient, in this case 5, then this is an equation of a vertical line. x = 5**Examples of Horizontal and Vertical Lines**The blue line means that whenever x is equal to any number, the y value will always be 5. Examples of coordinates would be: (-1, 5), (-8, 5), (7, 5), (100, 5) The value of y is always 5 which means it is a horizontal line. • y = 5Since y is equal to a coefficient, then this is an equation of a horizontal line. y = 5**Slope of a Line**• The formula for finding the slope of a line is: Caution: Be careful when substituting in the values of x and y into the formula. You may want to label your x and y variables in the coordinates so you do not mix them up. The “sub exponent 1” and “sub exponent 2” are very important.**Example for Finding the Slope of a Line**Labeling your points is very important. • Find the slope of the line with coordinates of (5, 7) and (9, 11).**You try it …..**How did you do? Click the mouse button to see the correct answer. But please do try it first! • Find the slope of the line with the points(-3, 2) and (2, -8)