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5.5 Standard Form: X-intercept: The point where the graph crosses the x-axis, ( y=0). Y-intercept: The point where the graph crosses the y-axis, (x=0). Linear Equation: is an equation that models a linear function. GOAL: Whenever we are given a graph we must be able to provide the equation of the function in Standard Form: The linear equation of the form: Ax + By = C where A, B and C are real whole numbers (no fractions) and A and B are not both zero. EX: What are the x- and y-intercepts of the graph of 5x – 6y = 60? SOLUTION: There are many ways to find this information depending on the form you are given, but if you are given the standard form (Ax+By=C), then you must plug in zero for the other variable. Finding the x-intercept: plug in zero for y 5x – 6y = 60 plug in y=0 5x – 6(0) = 60 5x = 60 x = 60/5 12 (12,0) is the point. Finding the y-intercept: plug in zero for x 5x – 6y = 60 plug in x=0 5(0) – 6y = 60 – 6y = 60 y=60/-6 -10 (0, -10) is the point Graph: − = X-intercept: (12, 0) 2 Y-intercept: ( 0, -10) -2 -2 2 YOU TRY IT: What are the x- and y-intercepts of the graph of 3x + 4y = 24? YOU TRY IT: (SOLUTION) Finding the x-intercept: plug in zero for y 3x + 4y = 24 3x + 4(0) = 24 plug in y=0 3x = 24 x = 24/3 8 (8,0) is the point. Finding the y-intercept: plug in zero for x 3x + 4y = 24 3(0) + 4y = 24 plug in X=0 4y = 24 y = 24/4 6 (0,6) is the point. Graph: X-intercept: (8, 0) Y-intercept: ( 0, 6) 3 + = Graphing Horizontal Lines Remember: x lines are vertical y lines are Horizontal X=3 y=-2 YOU TRY IT: What are the graphs of x = -1 and y = 5 YOU TRY IT: (SOLUTION) Remember: x lines are vertical y lines are Horizontal X = -1 y=5 TRANSFORMING TO STANDARD FORM If we are given an equation in slope-intercept from (y = mx +b), and the point-slope form (y – y1=m(x-x1)) we can rewrite the equations into standard form: Ax + By = C where A, B and C are real whole numbers (no fractions) and A and B are not both zero. EX: What are the standard forms of 1) y = - x + 5 and 2) y – 2 = - (x + 6) SOLUTION: 1) Using the slope-intercept from y = - x + 5 We must get rid of any fraction, no fractions allowed: y=- x+5 Inverse of dividing by 7 7y = - 3x + 35 Inverse subtraction 3x 7y + 3x= 35 Variables in order 3x + 7y = 35 Ax + By = C form. Graph: X-intercept: (11.7, 0) Y-intercept: (0, 5) Here we would use: y=- x+5 down 3, right 7 + = SOLUTION: 2) Using the point-slope from y-2 = - (x + 6) We must first distribute the slope y -2 = - x - 2 Distribute - We must then get rid of fractions Inverse of division by 3 3y - 6 = - x -6 (multiply everything by 3). 3y + X = -6 +6 Variables to left numbers to the right of equal sign. x + 3y = 0 Ax + By = C form. Graph: + = X-intercept: (0, 0) Y-intercept: (0, 0) We now use y=- x+0 USING STANDARD FORM AS MODEL In real-world situations we can write and use linear equations to obtain important information to help us find out what we can do with the resources we have. EX: In a video game, you earn 5 points for each jewel you find. You earn 2 points for each star you find. Write and graph an equation that represents the number of jewels and stars you must find to earn 250 points. What are three possible combinations of jewels and stars you can find that will earn you 250 points? SOLUTION: In a video game, you earn 5 points for each jewel you find. Let x = the jewels you find. You earn 2 points for each star you find. Let y = the starts you find. Write the equation for a total of 250 points: 5x + 2y = 250 Graph: 5 + = X-intercept: (50, 0) 250 Stars 225 200 Y-intercept: (0, 125) 175 150 125 100 75 50 25 25 50 75 100 125 Jewels Graph: Three points are: (0, 125) 0 Jewels, 125 Stars (25, 63) 25 Jewels, 62.5 Stars 225 200 Stars (25, 62.5) 25 Jewels, 62.5 Stars 250 175 150 125 100 75 50 25 25 50 75 100 125 Jewels VIDEOS: Graphs https://www.khanacademy.org/math/algebra/line ar-equations-and-inequalitie/point-slopeform/v/linear-equations-in-standard-form https://www.khanacademy.org/math/algebra/line ar-equations-and-inequalitie/point-slopeform/v/point-slope-and-standard-form CLASSWORK: Page 323-325 Problems: As many as needed to master the concept