1. The Game of Algebra or The Other Side of Arithmetic The Game of Algebra or The Other Side of Arithmetic © 2007 Herbert I. Gross by Herbert I. Gross & Richard A. Medeiros next Lesson 16

2. Solving Linear Equations Solving Linear Equations © 2007 Herbert I. Gross next Part 2 (Identities and Inconsistencies)

3. next © 2007 Herbert I. Gross In this Lesson we will look at cases in which an equation does not have any equilibrium points; and at the other extreme, we will look at cases in which there are infinitely many equilibrium points. next These cases occur whenever we have a linear equation in the form… mx + b = nx + c … in which m = n.

4. © 2007 Herbert I. Gross Candy To see how such a case can happen, let's return to our “candy store” example. Suppose the first candy store charges $4 per box and a $3 shipping and handling charge; while the second store charges $4 per box and a $5 shipping and handling charge. Then, no matter how many boxes of candy you want to buy, the second store will charge you $2 more than the first store.

5. next © 2007 Herbert I. Gross Let's now translate this situation into an equation. The cost of buying x boxes of candy at the first store, in dollars, is 4x + 3 while the cost of buying x boxes of candy at the second store is 4x + 5. 4x + 3 = 4x + 5 next Hence the equilibrium equation is…

6. © 2007 Herbert I. Gross In this case there are the same number of x's on each side of the equation. Hence, when we subtract 4x from each side, the x terms will cancel one another. That is, if we subtract 4x from both sides of the equation, we obtain the equivalent equation… 4x + 3 = 4x + 5 next 4x + 3 = 4x + 5 – 4x – 4x 3 = 5

7. © 2007 Herbert I. Gross Notice that the equation above is a numerical statement that does not involve a variable. 3 = 5 next It is simply a false statement. In other words, no matter what value we choose for x, 3 is never equal to 5.

8. © 2007 Herbert I. Gross Don’t confuse… 3 = 5 Caution next The equation x + 3 = 5 contains an x term, and we can solve the equation by subtracting 3 from each side to conclude that x = 2. … with the equation x + 3 = 5

9. © 2007 Herbert I. Gross In the equation 3 = 5, however, if we subtract 3 from each side, we wind up with 0 = 2, which is still a false statement. next One way to interpret 3 = 5 or 0 = 2 is that the right hand side of the equation will always be 2 more than the left hand side, no matter what value we choose for x. In still other words, x = 2 is a solution for the equation x + 3 = 5, but there is no number that is a solution for the equation 3 = 5.

10. © 2007 Herbert I. Gross In terms of the present illustration, in the equation 3 = 5, the right hand side is always 2 more than the left hand side. And since the right hand side represents the second store's price and the left hand side represents the first store's price, the above equation tells us that the second store's price is always $2 more than the first store's price.

11. next © 2007 Herbert I. Gross At the other extreme, there are equations for which every value of x is an equilibrium point. As a trivial example, suppose now that both candy stores charge $4 per box for the candy and $3 for shipping and handling charges. Then no matter how many boxes of candy you buy, it costs you the same amount regardless of which store you use. Again, in terms of a linear equation, the price each store charges, in dollars, is 4x + 3.

12. next © 2007 Herbert I. Gross Hence, the equilibrium equation is… 4x + 3 = 4x + 3 If we subtract 4x from both sides, we get… – 4x – 4x 3 = 3 which is a true statement no matter what value we choose for x. next

13. © 2007 Herbert I. Gross Both equations 3 = 5 and 3 = 3 contain no variables. They are called numerical equations (or numerical statements). A numerical equation must either be always true, or it must be always false. Note

14. next © 2007 Herbert I. Gross 4x + 3 = 4x +3 Most of us would recognize that … next … has every value of x as a solution because we can see that the two sides of the equation are identical. In fact, the equation 4x + 3 = 4x + 3 is called an identity.

15. © 2007 Herbert I. Gross next For example, the commutative property of arithmetic tells us that for every value of x… x + 1 = 1 + x The fact that equation is true for every value of x means that we may refer to the equation as being an identity. An equation is called an identity, if every value of the variable is a solution of the equation. Definition D

16. next © 2007 Herbert I. Gross However, it often happens in an identity that the two sides do not look alike until after we apply one or more rules of arithmetic. Therefore, we often have to simplify the linear expressions in the same way as we did in Lesson 15; the only difference being that this time we wind up with a numerical equation rather than an algebraic equation. Since we essentially do nothing else new in Lesson 16, we will limit further remarks to the solutions and commentary of the problems in the Lesson 16 exercise set.

17. next ♦ A relationship is called linear if the rate of change of the output with respect to the input is a constant. If we denote this constant by m, the input by x, the output by y, and the initial value of the output by b (that is, the value of y when x = 0), every linear relationship can be written in the form… However, before we do this, let's summarize the results we've obtained about linear relationships up until now. y = mx + b

18. © 2007 Herbert I. Gross next ♦ It isn't always obvious to see the “y = mx + b” form. Often we have to use the various rules of arithmetic to paraphrase the linear relationship into this form. If a relationship cannot be paraphrased into a “y = mx + b” form, the relationship is not linear. ♦ In summary, any linear expression (in which the variable is denoted by x) is an expression that can be paraphrased into the “mx + b” form.

19. © 2007 Herbert I. Gross next ♦ A linear equation is one in which both sides of the equation consist of linear expressions. ♦ In solving a linear equation, one, and only one, of three things must happen. Namely… (Case 1) The linear equation is satisfied by only one value of x. (Case 2) The linear equation is satisfied by no value of x. (Case 3) The linear equation is satisfied by every value of x.

20. © 2007 Herbert I. Gross next Case 1 Case 1 will happen whenever there are a different number of x's on one side of the equation than on the other. When this happens, we simply subtract the smaller number of x's from both sides of the equation, and proceed as before to solve for the specific value of x. In other words, Case 1 occurs in the solution of the equation mx + b = nx + c whenever m ≠ n.

21. © 2007 Herbert I. Gross next Case 2 and 3 Cases 2 and 3 occur whenever m = n. In this case, the equation mx + b = nx + c becomes… mx + b = mx +c And if we subtract mx from both sides of this equation, we get the numerical statement b = c which is either always true or always false. In other words, if b = c, the equation is an identity, and if b ≠ c, the equation has no solutions (in which case the equation is called inconsistent).

22. © 2007 Herbert I. Gross next Linear Equations and Straight lines Geometric interpretations can often shed light on discussions that might otherwise seem rather abstract. This applies to our discussion of how to solve linear equations. As a case in point, suppose we think in terms of the graphs of the equations rather than in terms of the equations themselves.

23. © 2007 Herbert I. Gross next For example, in terms of the Cartesian plane, the linear equation mx + b = nx + c determines the point at which the lines whose equations are y = mx + b and y = nx + c intersect. Recall that m and n are the rates of change of y with respect to x. Therefore, if m ≠ n, the two equations define lines that have different directions and hence are not parallel. In that case, the two lines will intersect at one and only one point.

24. © 2007 Herbert I. Gross next For example, suppose we want to solve the equation 4x +1 = 3x + 2 geometrically. We could begin by drawing the line y = 4x +1. (0,0) (1,5) y = 4x + 1 m = 4, (0,1) b = 1 next

25. © 2007 Herbert I. Gross next We could then continue by drawing the line y = 3x +2. (0,0) (1,5) y = 3x + 2 m = 3, (0,2) b = 2 next

26. © 2007 Herbert I. Gross next Drawn on the same grid, we see that the lines intersect at (1,5). y = 4x + 1 (0,0) (1,5) (0,1) next y = 3x + 2 (0,2)

27. © 2007 Herbert I. Gross next We can interpret the point (1,5) algebraically in the following way… Another Connection Starting with the equation 4x + 1 = 3x + 2, we subtract 3x from both sides to obtain x + 1 = 2; from which it follows that x = 1. When x = 1, both 4x + 1 and 3x + 2 equal 5. Thus, the point (1,5) may be used as an abbreviation for saying “x = 1 is the solution of the equation 4x + 1 = 3x + 2, and in this case both 4x + 1 and 3x + 2 equal 5”.

28. © 2007 Herbert I. Gross next However, if m = n, the two equations define lines that have the same directions (slope). This lead to two possibilities… ♦ either the two equations define the same line; which means every point that satisfies one equation will also satisfy the other equation. For example, if we multiply both sides of the equation x + y = 1 by 2, we obtain the equivalent equation 2x + 2y = 2. Thus, while the two equations are different, they define the same line; namely the line that is determined by the two points (0,1) and (1,0).

29. © 2007 Herbert I. Gross next ♦ the two equations define different but parallel lines; which means that no point on one line can also be on the other line. For example, the lines whose equations are y = 4x + 1 and y = 4x + 3 are different but parallel lines. Recall that when we solved the equation 4x + 1 = 4x + 3, we obtained the true statement 0 = 2. Geometrically, this means that the line y = 4x + 3 is point- by-point 2 units above the line y = 4x + 1.

30. © 2007 Herbert I. Gross y = 4x + 1 (0,0) (1,5) (0,1) next y = 4x + 3 (0,3) (1,7) More specifically, we draw the lines… Moreover… next …and we see that the lines are parallel.

31. © 2007 Herbert I. Gross (0,0) (1,5) (0,1) y = 4x + 1 y = 4x + 3 (0,3) (1,7) the vertical distance between the two lines is always 2 units. 2 2 next

32. © 2007 Herbert I. Gross next In theory a point has no thickness. However, the “dot” that we use to represent the point does have thickness. And since we may think of a line as being generated by a moving point, it means that any line we draw also has thickness. Points vs. Dots or Thickness Matters

33. © 2007 Herbert I. Gross next What this means is that when we represent equations by lines, the thickness of the lines often forces us to estimate the correct answer rather than to find the exact answer. By way of illustration, consider the line L 1 whose equation is… y = 7x + 3 …and suppose we want to find the (exact) point on L 1 whose y-coordinate is 15.

34. © 2007 Herbert I. Gross next Algebraically, we simply replace y by 15 in the equation y = 7x + 3 and obtain the equation… 15 = 7x + 3 If we subtract 3 from both sides of this equation and then divide both sides of the resulting equation by 7, we see that the x-coordinate of the point is given exactly by x = 12 / 7.

35. © 2007 Herbert I. Gross next However, if we were to rely solely on a geometric solution, we could begin by… ♦ first drawing the line y = 7x + 3, ♦ and then drawing the line y = 15. ♦ We would then label the point at which these two lines intersect as P(x,15). ♦ To determine the value of x, we would then draw the line L 2 that passes through P and is parallel to the y-axis. ♦ The desired value of x is the x-coordinate of the point at which this line crosses the x-axis.

36. © 2007 Herbert I. Gross next PictoriallyPictorially y x y = 15 y = 7x + 3 L2L2 L1L1 P = (x,15) next (?,0) next

37. © 2007 Herbert I. Gross next The problem is that the best we can do is estimate this point and conclude that the value of x is a “little more” than 1.5. So while the geometric model is often a good way to help us internalize what is happening algebraically, it is the algebraic method that gives us the exact answer. However, there are times when an exact algebraic solution doesn't exist and in such cases the graph(s) of the equation(s) is extremely useful in helping us obtain a reasonable approximation for the exact answer to the problem.

38. © 2007 Herbert I. Gross next Summary of Lessons 15 and 16 Lest we lose sight of the forest because of the trees, remember that these two lessons were concerned with solving linear equations of the form… mx + b = nx + c What we showed (basically by algebra but illustrated in terms of coordinate geometry) was that for each such equation, one and only one of the following solutions occurrs…

39. © 2007 Herbert I. Gross next ♦ If m ≠ n, there is one and only one value of x for which the equation is a true statement. ♦ If m = n but b ≠ c, there is no value of x for which the equation is a true statement. ♦ If m = n and it is also true that b = c, the expressions on either side of the equal sign are equivalent and in this case the equation becomes a true statement for every value of x.