1. Steps to Solve Word Problems 1.Read the problem carefully. 2.Get rid of clutter 3.Identify key variables (unknowns). 4.Eliminate unneeded variables. 5.Use the text of the problem to write equations. 6.Solve the equation. 7.Find the remaining variables.

2. Example of Word Problem If 4 apples and 2 oranges equals $1 and 2 apples and 3 orange equals $0.70, how much does each apple and each orange cost? There are no quantity discounts.

3. What do we know? If 4 apples and 2 oranges equals $1 and 2 apples and 3 orange equals $0.70, how much does each apple and each orange cost? There are no quantity discounts. Step 1: Now you need to identify the unknowns. An unknown is a quantity that your problem requires you to find out, or the quantity that is necessary to find out in order to obtain the solution. What are the unknowns in this problem? The price of 1 apple and 1 orange

4. What do we know? If 4 apples and 2 oranges equals $1 and 2 apples and 3 orange equals $0.70, how much does each apple and each orange cost? There are no quantity discounts. Step 2: Now you need to find out the equation needed to solve the problem. The problem basically gives you the equations, you just need to write them in mathematical form. 4x + 2y = 1, 2x + 3y = 0.70

5. What do we know? If 4 apples and 2 oranges equals $1 and 2 apples and 3 orange equals $0.70, how much does each apple and each orange cost? There are no quantity discounts. Step 2: Now you need to find out the equation needed to solve the problem. The problem basically gives you the equations, you just need to write them in mathematical form. 4x + 2y = 1, 2x + 3y = 0.70

6. Now you Try The senior class at your high school has it prom at a banquet facility. The banquet facility $15.95 per person for a dinner buffet and $400 to rent the banquet hall for an evening. The class paid the banquet facility a total of $2633 for the dinner buffet and use of the banquet hall. How many people attended the prom? 140 people

7. Algebra 1 Ch 4.2 – Graphing Linear Equations

8. Objective Students will graph linear equations using a table. Students will graph linear equations using a table. Students will graph horizontal and vertical lines Students will graph horizontal and vertical lines

9. Vocabulary A linear equation in two variables is an equation in which the variables appear in separate terms and neither variable contains an exponent other than 1. A linear equation in two variables is an equation in which the variables appear in separate terms and neither variable contains an exponent other than 1. The solution to linear equations are ordered pairs which makes the equation true. The solution to linear equations are ordered pairs which makes the equation true. The graph of an equation in x and y is the set of all points (x, y) that are solutions of the equation. The graph of an equation in x and y is the set of all points (x, y) that are solutions of the equation.

10. Example #1 This is an example of a linear equation This is an example of a linear equation y = x + 8 All linear equations are functions. That is the value of y (output) is determined by the value of x (input) All linear equations are functions. That is the value of y (output) is determined by the value of x (input) The linear equation in two variables can also be called the rule. The linear equation in two variables can also be called the rule.

11. Solution Any ordered pair of numbers that makes a linear equation true. (9,0) IS ONE SOLUTION FOR Y = X - 9

12. Examples of Solutions 2y + x = 4 2y + x = 4 Is (-2,3) a solutions to this equation? Is (-2,3) a solutions to this equation?Solution!

13. Examples of solutions

14. Graphing Since the results of a linear equations can be expressed as ordered pairs, the linear equation can be graphed. Since the results of a linear equations can be expressed as ordered pairs, the linear equation can be graphed. When a linear equation is graphed, all points on the line represent the solution set of the linear equation. When a linear equation is graphed, all points on the line represent the solution set of the linear equation. There are a number of ways to find the solution to a linear equation…for today’s lesson we will look at creating a table of the solutions… There are a number of ways to find the solution to a linear equation…for today’s lesson we will look at creating a table of the solutions…

15. Tables To create a table of solutions to a linear equation do the following: To create a table of solutions to a linear equation do the following: 1. 1. Choose a minimum of 3 values for x 2. 2. Substitute the values of x into the linear equation 3. 3. Simplify to find the value of y 4. 4. Write the solutions as ordered pairs Let’s look at an example… Let’s look at an example…

16. Example #2 y = 2x – 1 x y (x, y) 0 y = 2(0) – 1 - 1 (0, - 1) 1 y = 2(1) – 1 1 (1, 1) 2 y = 2(2) – 1 3 (2, 3) 1. Choose a minimum of 3 values for x 2. Substitute the value of x into the equation 3. Simplify to determine the value of y 4. Write as an ordered pair

17. Graphing Use the ordered pair from the table to graph the linear equation. Use the ordered pair from the table to graph the linear equation. Again…when graphing the result should be a straight line… Again…when graphing the result should be a straight line… Any point (ordered pair) on that line will be a solution to the linear equation… Any point (ordered pair) on that line will be a solution to the linear equation…

18. (x, y) (0, - 1) (1, 1) (2, 3) x y (0, -1) (1,1) (2,3) y = 2x – 1

19. Linear Equations All linear equations can be written in the form: All linear equations can be written in the form: Ax + By = C This form is called the standard form of an equation. This form is called the standard form of an equation. At this level you are required to know and be able to manipulate this form of an equation At this level you are required to know and be able to manipulate this form of an equation

20. Standard Form Ax + By = C In the standard form of an equation: In the standard form of an equation: A is the coefficient of x A is the coefficient of x B is the coefficient of y B is the coefficient of y C represents the constant C represents the constant We talked about coefficients and constants in a previous lesson We talked about coefficients and constants in a previous lesson

21. Example #3 The equation 3x – 4y = 12 is an example of an equation written in standard form. The equation 3x – 4y = 12 is an example of an equation written in standard form. As we have done in a previous lesson, we can write the equation in function form by transforming the equation as follows: As we have done in a previous lesson, we can write the equation in function form by transforming the equation as follows: 3x – 4y = 12 -3x -3x – 4y = -3x + 12 – 4y = -3x + 12 – 4 y = - ¾ x + 3 Standard Form Function Form

22. Horizontal Lines In the standard form of an equation Ax + By = C, When A=0 the equation reduces to By = C and the graph will be a horizontal line. In the standard form of an equation Ax + By = C, When A=0 the equation reduces to By = C and the graph will be a horizontal line. We often see this illustrated as the equation y = b. We often see this illustrated as the equation y = b. In this instance, the equation has no x-value and the y-value is always the same number so that when the y-value is graphed a horizontal line is produced. In this instance, the equation has no x-value and the y-value is always the same number so that when the y-value is graphed a horizontal line is produced.

23. Example #4 – Horizontal Line Graph the equation y=2 Graph the equation y=2 In this instance there is no x-value. All the y-values = 2 In this instance there is no x-value. All the y-values = 2 To plot this line, starting at 0, go up 2 spaces on the y-axis and draw a horizontal line (as shown in the next slide) To plot this line, starting at 0, go up 2 spaces on the y-axis and draw a horizontal line (as shown in the next slide)

24. Example #4 (Continued) x yy=2 y=2

25. Comments Notice that when you graph the line, the line is perpendicular to the y-axis. Notice that when you graph the line, the line is perpendicular to the y-axis. A common error that students make when graphing an equation like y=2 is that they draw the line parallel to the y-axis. That is incorrect! A common error that students make when graphing an equation like y=2 is that they draw the line parallel to the y-axis. That is incorrect! A way to avoid this error is to actually plot the point before you draw the line. A way to avoid this error is to actually plot the point before you draw the line.

26. Vertical Lines In the standard form of an equation Ax + By = C, When B=0 the equation reduces to Ax = C and the graph will be a vertical line. In the standard form of an equation Ax + By = C, When B=0 the equation reduces to Ax = C and the graph will be a vertical line. We often see this illustrated as the equation x = a We often see this illustrated as the equation x = a In this instance, the equation has no y-value and the x-value is always the same number so that when the x-value is graphed a vertical line is produced In this instance, the equation has no y-value and the x-value is always the same number so that when the x-value is graphed a vertical line is produced

27. Example #5 – Vertical Line Graph the equation x = -3 Graph the equation x = -3 In this instance there is no y-value. All the x-values = -3 In this instance there is no y-value. All the x-values = -3 To plot this line, starting at 0, go 3 spaces to the left on the x-axis and draw a vertical line (as shown in the next slide) To plot this line, starting at 0, go 3 spaces to the left on the x-axis and draw a vertical line (as shown in the next slide)

28. Example #5(Continued) x yx=-3 x = -3

29. Comments Notice that when you graph the line, the line is perpendicular to the x-axis. Notice that when you graph the line, the line is perpendicular to the x-axis. A common error that students make when graphing an equation like x=-3 is that they draw the line parallel to the x-axis. That is incorrect! A common error that students make when graphing an equation like x=-3 is that they draw the line parallel to the x-axis. That is incorrect! A way to avoid this error is to actually plot the point before you draw the line. A way to avoid this error is to actually plot the point before you draw the line.

30. Your Turn Find 3 different ordered pairs that are the solutions to the equation Find 3 different ordered pairs that are the solutions to the equation 1. y = 3x – 5 2. y = -2x – 6 3. y = ½ (4 – 2x) 4. y = 4( ½ x – 1)

31. Your Turn Rewrite the equation in function form Rewrite the equation in function form 5. 2x + 3y = 6 6. 5x + 5y = 19 Create a table of values & graph the linear equation Create a table of values & graph the linear equation 7. y = -x + 4 8. y= -(3 – x) 9. x = 9 10. y = -1

32. Your Turn Solutions 1. (-1,-8), (0,-5),(1, -2) 2. (-1,-4),(0,-6),(1,-8) 3. (-1,3),(0,2),(1,1) 4. (-1,-6),(0,-4),(1,-2) 5. y = -2/3x + 2 6. y = -x + 19/5 You should have a table with a minimum of 3 values. When plotting the line the following should be true: You should have a table with a minimum of 3 values. When plotting the line the following should be true: 7. Your graph should cross the y-axis at +4 8. Your graph should cross the y-axis at -3 9. You should have a vertical line at the point x = 9 10. You should have a horizontal line at the point y= -1

33. Homework Section 4.2 pg.219-220 Section 4.2 pg.219-220 #16, 11-21(ODD), 23-25, 27, 36