Section 1.2. Graphing Linear Equations
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1 Graphing Linear Equations
2 Definition of Solution, Satisfy, and Solution Set Definition of Solution, Satisfy, and Solution Set Consider the equation y = 2x 5. Let s find y when x = 3. y = 2x 5 Original Equation. y = 23 5 Substitute 3 for x. = 6 5 Multiply before subtracting. = 1 Subtract. So, y = 1 when x = 3, which cab be represented by the ordered pair 3,1. Slide 2
3 Definition of Solution, Satisfy, and Solution Set Definition Definition of Solution, Satisfy, and Solution Set For an ordered pair ( ab,,) we write the value of the independent variable in the first (left) position and the value of the dependent variable in the second (right) position. The numbers a and b are called coordinates. For ( 3,1 ), the x-coordinate is 3 and the y-coordinate is 1. Slide 3
4 Definition of Solution, Satisfy, and Solution Set Definition of Solution, Satisfy, and Solution Set The equation y = 2x 5 becomes a true statement when we substitute 3 for x- coordinate and 1 for y-coordinate. y?? = 2x 5 1= = 1 true Original Equation. Substitute 3 for x and 1 for y. Slide 4
5 Definition of Solution, Satisfy, and Solution Set Definition Definition of Solution, Satisfy, and Solution Set An ordered pair ( ab, ) is a solution of an equation in terms of x and y if the equation becomes a true statement when a is substituted for x and b is substituted for y. We say ( ab, ) satisfies the equation. The solution set of the equation is the set of all solution of the equation. Slide 5
6 Example Graphing an Equation Definition of Solution, Satisfy, and Solution Set Find five solutions to the equation y = 2x+ 1, and plot them in the coordinate system (on the right). Slide 6
7 Solution Graphing an Equation Definition of Solution, Satisfy, and Solution Set We begin be arbitrarily choosing the values 0, 1, and 2 to substitute for x: ( ) y = y = y = = 0+ 1 = 2+ 1 = 4+ 1 = 1 = 1 = 3 Solution: 0,1 Solution: 1, 1 Solution: 2, 3 ( ) The ordered pairs 2,5 and 1, 3 are also solutions. ( ) Slide 7
8 Graphing an Equation Definition of Solution, Satisfy, and Solution Set Solution Continued Create a table of solutions x y Plot the solutions Points form a linear line. Slide 8
9 Graphing an Equation Definition of Solution, Satisfy, and Solution Set Every point on the line is a solution to the equation y = 2x+ 1 Slide 9
10 Graphing an Equation Definition of Solution, Satisfy, and Solution Set The point ( 3, 5) lies on the line Should satisfy the equations Whereas ( 2,4) is not on the line Thus should not satisfy the equation y = 2x+ 1 Slide 10
11 Graphing an Equation Definition of Solution, Satisfy, and Solution Set y = 2x+ 1? 4= 22+ 1? 4= 3 false Original Equation. Substitute 2 for x and 4 for y. The ( 2,4) is not a solution to the equation Slide 11
12 Graphing an Equation Definition of Solution, Satisfy, and Solution Set Calculator Use ZDecimal on a graphing calculator. To enter y = 2x+ 1, press ( ) 2 X,T,ϴ,n + 1. The key is used for subtraction, and the key. ( ) is used for negative numbers as well as taking the opposite. Slide 12
13 Definition Definition: Graph Definition of Solution, Satisfy, and Solution Set The graph of an equation in two variables is the set of points that correspond to all solutions of the equation. In the last example we found that the equation. y = 2x+ 1, is a line. Notice that the equation. y = 2x+ 1, is of the form y = mx + b (where m = 2 and b = 1). Slide 13
14 Graphs of Linear Equations Equations of the form y = mx + b If an equation can be put into the form y = mx + b where m and b are constants, then the graph of the equation is a line. Example Graphs of Linear Equations What is m and b for the equations 3 y = x 2, y = 3x and y = 3? 2 Slide 14
15 Definition Graphing Linear Equations 3 3 y = x 2is of the form y = mx + b: m = and b = y = 2xis of the form y = mx + b because we write the equation as y = 2x+ 0 (so m = 2 and b = 0). y = 3 is of the form y = mx + bbecause we write the equation as y = 0x+ 3(so m = 0 and b = 3). Example Graphs of Linear Equations Sketch the graph of the equation 30x 6y+ 5 = 0.
16 Graphing Linear Equations Graphs of Linear Equations Solution First we solve for y 30x 6y+ 12 = 0 30x 6y = 12 6y = 30x y = x y = 5x 2 Original Equation. Subtract 12 from both sides. Subtract 30x from both sides. Divide both sides by 6. Simplify. Slide 16
17 Graphing Linear Equations Solution Continued Graphs of Linear Equations. y = 5x 2is of the form y = mx + b The graph of the equation is a line Find 2 points of the line Plot the two points Sketch the line Find a third point Verify that the third point lies on the line Slide 17
18 Graphing Linear Equations Solution Continued Table of solutions Graphs of Linear Equations x y = = = 1 Slide 18
19 Graphing Linear Equations Graphing Calculator Enter 5x 2 for y 1. Use Zstandard followed by Zsquare. The graph is correct assuming that y was isolated correctly. Graphs of Linear Equations Slide 19
20 Using the Distributive Law to Help Graph a Linear Equation Example Sketch the graph of Solution Graphs of Linear Equations 32y 5= 2x 3 8. x Use the distributive property on the left-hand side. Collect like terms. Isolate y. Slide 20
21 Using the Distributive Law to Help Graph a Linear Equation Graphs of Linear Equations Solution Continued 32y 5= 2x 3 8x 6y 15 = 6x 3 6y = 6x y = x y = x+ 2 Original equation Distributive property Add 15 to both sides. Divide both sides by 6. Simplify. Slide 21
22 Using the Distributive Law to Help Graph a Linear Equation Solution Continued Table of solutions Graphs of Linear Equations x y = = = 0 Slide 22
23 Graphing an Equation That Contains Fractions Example Sketch a graph of y Solution Graphs of Linear Equations 1 = x 1. 2 Avoid fraction values for y Use even values for x Slide 23
24 Graphing an Equation That Contains Fractions Solution Continued Table of solutions x y 0 2 Graphs of Linear Equations 1 ( 0 ) 1 = ( 2 ) 1 = ( 4 ) 1 = 1 2 Slide 24
25 Graphing an Equation That Contains Fractions Graphing Calculator Graphs of Linear Equations Use Zdecimal to verify the solution. Slide 25
26 Property Finding Intercepts of a Graph Sometimes we find intercepts to graph a line. x-intercept is on the y-axis, so y = 0 y-intercepts in on the x-axis, so x = 0 Directions For an equation containing the variables x and y x-intercept: Substitute y = 0 and solve for x y-intercept: Substitute x = 0 and solve for y Slide 26
27 Using Intercepts to Sketch a Graph Example Use intercepts to sketch a graph of y = 2x+ 4. Solution Finding Intercepts of a Graph x-intercept: Set y = 0. 0= 2x + 4 4= 2x Substitute 0 for y. Subtract both sides by = x Divide both sides by = x Simplify. Slide 27
28 Using Intercepts to Sketch a Graph Solution Continued y-intercept: Set x = 0. Finding Intercepts of a Graph y y = = 4 Set x =0. Simplify. So, the x-intercept is ( 2,0) and y-intercept is 0,4. Slide 28
29 Using Intercepts to Sketch a Graph Graphing Calculator Use ZStandard followed by ZSquare. Finding Intercepts of a Graph Use zero to verify the x-intercept. Use TRACE to verify the y-intercept. Slide 29
30 Graph the equation of x Example Solution y Graphing a Vertical Line Finding Intercepts of a Graph x = 3. Notice that the values of x must be 3, but y can have any value. Some solutions are listed to the left. Slide 30
31 Graphing a Horizontal Line Example Graph the equation of Solution x y Vertical and Horizontal Lines y = 5. Notice that the values of y must be 5, but x can have any value. Some solutions are listed to the left. Slide 31
32 Graphing a Horizontal Line Graphing Calculator Vertical and Horizontal Lines Use ZStandard to verify the graph. Slide 32
33 Vertical and Horizontal Line Property Property If a and b are constants: Vertical and Horizontal Lines An equation that can be put into the form x= a. has a vertical line as its graph An equation that can be put into the form y = b.has a horizontal line as its graph Slide 33
34 Vertical and Horizontal Line Property Property Vertical and Horizontal Lines In an equation can be put into either form y = mx + b or x = a where m, a, and b are constants, then the graph of the equation is a line. We call such an equation a linear equation in two variables. Slide 34
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