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6.2 Graphing Linear Equations Equations with only variable and with no exponents have only one solution. E.g. 2x – 1 = -5 The only solution is x=-2 However, linear equations, which are equations with two variables and no exponents, have an infinite number of solutions. How about 2x - 3y = 6 ? What values of x and y make this equation true? Well, you could have x=- and y = -2 [2(0) – 3(-2) = 6] Or you could have x=-3 and y = -4 [2(-3) – 3(-4) = -6+12 = 6] And so on and so on. There are too many to name. We can state the solution by graphing it. How do you graph an equation like 2x – 3y = 6? 4 y 3 2 1 0 -4 -3 -2 -1 -1 0 -2 -3 -4 -5 x 1 2 3 4 Linear equations have a dependent variable and an independent variable. In general, x is the independent variable and y is the dependent variable. When a linear equation is re-written so that y is on one side and x is on the other, the value of y depends on the value chosen for x. How to graph a linear equation: Step 1) Re-arrange the equation so that y is isolated. 2x - 3y = 6 +2x +2x -3y = 2x + 6 (even though we added 2x to 6 it is “proper” to place the variable term before the constant term.) Now divide both sides by -3 to get y by itself. − 3y − 2x 6 = + −3 −3 −3 2 y = x−2 3 Step 2) Set up a table of ordered pairs and choose x’s, then find out what the y’s need to be. You only need two ordered pairs to draw a line, but three helps it to be more accurate. When choosing x’s, try to pick ones that will come out with an integer x y=⅔x-2 (x,y) when multiplied by the coefficient -3 =⅔(-3)-2=-2-2=-4 (-3,-4) in your linear equation. Now that you have three ordered pairs, just plot them on a graph and connect the dots! 0 =⅔(0) -2=-2 (0,-2) 3 =⅔(3)-2=2-2=0 (3,0) Sometimes it’s hard to choose x’s that will give you a nice integer for y. An alternative method is using the slope and intercept of the line. Graph 2y – x = 2 Get y by itself by first adding x to both sides 2y - x + x = 2 + x 2y = x + 2 Divide both sides by 2 2y x 2 = + 2 2 2 1 y = x +1 2 Once y is isolated, it is now in what is called the “slope-intercept form.” y = mx + b, where m represents something called the slope and b represents the y-intercept. The slope of the line tells you how much it is slanted. The y-intercept is the point where the line crosses the y-axis. At that point x=0 and y is the Run 2 units constant. y 2 In this case, the slope is ½ and the y-intercept is 1 [or more specifically (0,1)] A slope of ½ means that from any point on the line, the line rises vertically 1 unit and “runs” horizontally 2 units. Rise 1 unit 1 x 0 -2 -1 0 1 -1 -2 We know right away that one point on the line is the y-intercept: (0,1). So we can start at that point and rise 1 unit and then run 2 units to land at another point on the line (2,2). 2 Slopes tell you immediately about the steepness and direction of a line. A slope that is positive points up and to the right A slope that is negative points down and to the right. If the rise is negative, the vertical rise in a downward direction. If the run is negative, the horizontal “run” is to the left. −2 2 = 1 −1 down 2 units up 2 unit = = same " slant" right 1 unit left 1 units A slope of - 2 = A slope whose absolute value is a fraction less than 1 is closer to being horizontal. A slope whose absolute value is greater than one is steeper and closer to being vertical Graph y = 3x – 2 This is already in slope-intercept form. The slope is ____ The y-intercept is (0,__) y Run 1 unit 6 5 4 Rise 3 units 3 2 1 0 -6 -5 -4 -3 -2 -1 x 0 -1 -2 -3 -4 -5 -6 1 2 3 4 5 6