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Section 1.2 Graphing Linear Equations Definition of Solution, Satisfy, and Solution Set D e f i n i t i o n o f S o l u t i o n , S a t i s f y, a n d S o l u t i o n S e t Consider the equation y 2 x 5. Let’s find y when x 3. y 2x 5 Original Equation. y 2 3 5 Substitute 3 for x. 65 Multiply before subtracting. 1 Subtract. So, y 1 when x 3, which cab be represented by the ordered pair 3,1 . Section 1.2 Lehmann, Intermediate Algebra, 3ed Slide 2 Definition of Solution, Satisfy, and Solution Set D e f i n i t i o n o f S o l u t i o n , S a t i s f y, a n d S o l u t i o n S e t Definition For an ordered pair a, b, 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. Section 1.2 Lehmann, Intermediate Algebra, 3ed Slide 3 Definition of Solution, Satisfy, and Solution Set D e f i n i t i o n o f S o l u t i o n , S a t i s f y, a n d S o l u t i o n S e t The equation y 2 x 5 becomes a true statement when we substitute 3 for xcoordinate and 1 for y-coordinate. y 2x 5 Original Equation. ? 1 2 3 5 Substitute 3 for x and 1 for y. ? 1 1 true Section 1.2 Lehmann, Intermediate Algebra, 3ed Slide 4 Definition of Solution, Satisfy, and Solution Set D e f i n i t i o n o f S o l u t i o n , S a t i s f y, a n d S o l u t i o n S e t Definition • An ordered pair a, b 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 a, b satisfies the equation. • The solution set of the equation is the set of all solution of the equation. Section 1.2 Lehmann, Intermediate Algebra, 3ed Slide 5 Graphing an Equation D e f i n i t i o n o f S o l u t i o n , S a t i s f y, a n d S o l u t i o n S e t Example Find five solutions to the equation y 2 x 1, and plot them in the coordinate system (on the right). Section 1.2 Lehmann, Intermediate Algebra, 3ed Slide 6 Graphing an Equation D e f i n i t i o n o f S o l u t i o n , S a t i s f y, a n d S o l u t i o n S e t Solution We begin be arbitrarily choosing the values 0, 1, and 2 to substitute for x: y 2 0 1 0 1 1 y 2 1 1 y 2 2 1 2 1 1 4 1 3 Solution: 0,1 Solution: 1, 1 Solution: 2, 3 The ordered pairs 2,5 and 1,3 are also solutions. Section 1.2 Lehmann, Intermediate Algebra, 3ed Slide 7 Graphing an Equation D e f i n i t i o n o f S o l u t i o n , S a t i s f y, a n d S o l u t i o n S e t Solution Continued • Create a table of solutions x -2 -1 0 1 2 Section 1.2 y 5 3 1 -1 -3 • Plot the solutions • Points form a linear line. Lehmann, Intermediate Algebra, 3ed Slide 8 Graphing an Equation D e f i n i t i o n o f S o l u t i o n , S a t i s f y, a n d S o l u t i o n S e t • Every point on the line is a solution to the equation y 2 x 1 Section 1.2 Lehmann, Intermediate Algebra, 3ed Slide 9 Graphing an Equation D e f i n i t i o n o f S o l u t i o n , S a t i s f y, a n d S o l u t i o n S e t • 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 2 x 1 Section 1.2 Lehmann, Intermediate Algebra, 3ed Slide 10 Graphing an Equation D e f i n i t i o n o f S o l u t i o n , S a t i s f y, a n d S o l u t i o n S e t y 2 x 1 Original Equation. ? 4 2 2 1 Substitute 2 for x and 4 for y. ? 4 3 false • The 2, 4 is not a solution to the equation Section 1.2 Lehmann, Intermediate Algebra, 3ed Slide 11 Graphing an Equation D e f i n i t i o n o f S o l u t i o n , S a t i s f y, a n d S o l u t i o n S e t Calculator Use ZDecimal on a graphing calculator. • To enter y 2 x 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. Section 1.2 Lehmann, Intermediate Algebra, 3ed Slide 12 Definition: Graph D e f i n i t i o n o f S o l u t i o n , S a t i s f y, a n d S o l u t i o n S e t Definition 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 2 x 1, is a line. Notice that the equation y 2 x 1, is of the form y mx b (where m 2 and b 1). Section 1.2 Lehmann, Intermediate Algebra, 3ed . . Slide 13 Graphs of Linear Equations 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 What is m and b for the equations 3 y x 2, y 3x and y 3? 2 Section 1.2 Lehmann, Intermediate Algebra, 3ed Slide 14 Graphing Linear Equations Graphs of Linear Equations Definition 3 3 y x 2 is of the form y mx b: m and b 2 2 2 y 2 x is of the form y mx b because we write the equation as y 2 x 0 (so m 2 and b 0). y 3 is of the form y mx bbecause we write the equation as y 0 x 3 (so m 0 and b 3). Example Sketch the graph of the equation 30 x 6 y 5 0. Lehmann, Intermediate Algebra, 3ed Graphing Linear Equations Graphs of Linear Equations Solution First we solve for y Original Equation. 30 x 6 y 12 0 Subtract 12 from both sides. 30 x 6 y 12 6 y 30 x 12 Subtract 30x from both sides. 6 30 12 Divide both sides by 6. y x 6 6 6 y 5 x 2 Simplify. Section 1.2 Lehmann, Intermediate Algebra, 3ed Slide 16 Graphing Linear Equations Graphs of Linear Equations Solution Continued • . y 5 x 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 Section 1.2 Lehmann, Intermediate Algebra, 3ed Slide 17 Graphing Linear Equations Graphs of Linear Equations Solution Continued Table of solutions x y 0 2 0 3 3 1 2 1 3 1 2 2 2 3 1 Section 1.2 Lehmann, Intermediate Algebra, 3ed Slide 18 Graphing Linear Equations Graphs of Linear Equations Graphing Calculator • Enter 5x 2 for y1. • Use Zstandard followed by Zsquare. • The graph is correct assuming that y was isolated correctly. Section 1.2 Lehmann, Intermediate Algebra, 3ed Slide 19 Using the Distributive Law to Help Graph a Linear Equation Graphs of Linear Equations Example Sketch the graph of 3 2 y 5 2 x 3 8 x. Solution • Use the distributive property on the left-hand side. • Collect like terms. • Isolate y. Section 1.2 Lehmann, Intermediate Algebra, 3ed Slide 20 Using the Distributive Law to Help Graph a Linear Equation Graphs of Linear Equations Solution Continued 3 2 y 5 2 x 3 8 x Original equation Distributive property 6 y 15 6 x 3 6 y 15 15 6 x 3 15 Add 15 to both sides. 6 6 12 Divide both sides by 6. y x 6 6 6 Simplify. y x 2 Section 1.2 Lehmann, Intermediate Algebra, 3ed Slide 21 Using the Distributive Law to Help Graph a Linear Equation Graphs of Linear Equations Solution Continued Table of solutions x y 0 0 2 2 1 1 2 1 2 2 2 0 Section 1.2 Lehmann, Intermediate Algebra, 3ed Slide 22 Graphing an Equation That Contains Fractions Graphs of Linear Equations Example 1 Sketch a graph of y x 1. 2 Solution • Avoid fraction values for y • Use even values for x Section 1.2 Lehmann, Intermediate Algebra, 3ed Slide 23 Graphing an Equation That Contains Fractions Graphs of Linear Equations Solution Continued Table of solutions x 0 2 4 Section 1.2 y 1 0 1 1 2 1 2 1 0 2 1 4 1 1 2 Lehmann, Intermediate Algebra, 3ed Slide 24 Graphing an Equation That Contains Fractions Graphs of Linear Equations Graphing Calculator Use Zdecimal to verify the solution. Section 1.2 Lehmann, Intermediate Algebra, 3ed Slide 25 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 Section 1.2 Lehmann, Intermediate Algebra, 3ed Slide 26 Using Intercepts to Sketch a Graph Finding Intercepts of a Graph Example Use intercepts to sketch a graph of y 2 x 4. Solution x-intercept: Set y = 0. 0 2 x 4 Substitute 0 for y. 4 2 x Subtract both sides by 4. 4 2 Divide both sides by - 2. x 2 2 Simplify. 2x Section 1.2 Lehmann, Intermediate Algebra, 3ed Slide 27 Using Intercepts to Sketch a Graph Finding Intercepts of a Graph Solution Continued y-intercept: Set x = 0. y 2 0 4 Set x = 0. Simplify. y4 So, the x-intercept is 2,0 and y-intercept is 0, 4 . Section 1.2 Lehmann, Intermediate Algebra, 3ed Slide 28 Using Intercepts to Sketch a Graph Finding Intercepts of a Graph Graphing Calculator Use ZStandard followed by ZSquare. Use TRACE to verify the y-intercept. Use “zero” to verify the x-intercept. Section 1.2 Lehmann, Intermediate Algebra, 3ed Slide 29 Graphing a Vertical Line Finding Intercepts of a Graph Example Graph the equation of x 3. Solution x y 3 3 3 3 3 Section 1.2 5 3 1 -1 -3 Notice that the values of x must be 3, but y can have any value. Some solutions are listed to the left. Lehmann, Intermediate Algebra, 3ed Slide 30 Graphing a Horizontal Line Ve r t i c a l a n d H o r i z o n t a l L i n e s Example Graph the equation of y 5. Solution x y –2 –1 0 1 2 Section 1.2 –5 –5 –5 –5 –5 Notice that the values of y must be –5, but x can have any value. Some solutions are listed to the left. Lehmann, Intermediate Algebra, 3ed Slide 31 Graphing a Horizontal Line Ve r t i c a l a n d H o r i z o n t a l L i n e s Graphing Calculator Use ZStandard to verify the graph. Section 1.2 Lehmann, Intermediate Algebra, 3ed Slide 32 Vertical and Horizontal Line Property Ve r t i c a l a n d H o r i z o n t a l L i n e s Property If a and b are constants: • 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 Section 1.2 Lehmann, Intermediate Algebra, 3ed Slide 33 Vertical and Horizontal Line Property Ve r t i c a l a n d H o r i z o n t a l L i n e s Property In an equation can be put into either form y mx b or xa 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. Section 1.2 Lehmann, Intermediate Algebra, 3ed Slide 34