key stone problem… key stone problem… next set 21 © 2007 herbert i. gross

Key Stone Problem…Key Stone Problem…

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Set 21© 2007 Herbert I. Gross

You will soon be assigned problems to test whether you have internalized the material

in Lesson 21 of our algebra course. The Keystone Illustration below is a

prototype of the problems you’ll be doing. Work out the problems on your own.

Afterwards, study the detailed solutions we’ve provided. In particular, notice that several different ways are presented that could be used to solve each problem.

Instructions for the Keystone Problem

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© 2007 Herbert I. Gross

As a teacher/trainer, it is important for you to understand and be able to respond

in different ways to the different ways individual students learn. The more ways

you are ready to explain a problem, the better the chances are that the students

will come to understand.

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Two numbers are defined implicitly by the information that (1) the sum of twice the first number plus the second number is 13, and (2) the sum of 3 times the first number plus twice the second number

is 21. If we use x to denote the first number and y to denote the second

number, in explicit form, find the value of x and y.

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Keystone Problem for Lesson 21Keystone Problem for Lesson 21next

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Let’s begin by translating the two constraints into algebraic equations.

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If x denotes the first number, then 2x denotes twice the first number; and if y denotes the second number then 2x + y denotes the sum of twice the first number plus the second number. Since we are told that this sum is 13, our first algebraic equation is…

2x + y = 13

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SolutionSolution

Again letting x represent the first number and y represent the second number; 3x represents 3 times the first number, 2y

represents twice the second number and 3x + 2y represents the sum of 3 times the

first number plus twice the second number.Since we are told that this sum is 21, our second algebraic equation is…

3x + 2y = 21

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SolutionSolution

Since x and y must satisfy both equations they must be the solution of the linear system…

3x + 2y = 21 2x + y = 13

To solve our linear system we use our rules to eliminate either x or y. If we want to eliminate x, one way to do it is to multiply the top equation by -3,

-6x + -4y = -42 -6x + -3y = -39

and the bottom equation by 2 to obtain the equivalent system…

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SolutionSolution

If we now add the top and bottom equations in our system the terms

involving x “cancel”, and we see that y = 3.

If we then replace y by 3 in the equation 2x + y = 13, we see that 2x + 3 = 13,

which in turn means that 2x = 10 or x = 5.

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-6x + -4y = -42 -6x + -3y = -39

0 + y = 3

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SolutionSolution

As a final check, in the equation 3x + 2y = 21, we replace x by 5 and y by 3,

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3x + 2y = 21 2x + y = 13

and verify that it is a true statement.

3 x + 2 y = 21(5) (3)

15 + 6 = 21

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Notes on our Solution

A more formal (but admittedly a much longer) way to present the above solution is in terms of continuing the process of

replacing one system of equations by an equivalent system of equations.

For example, in our solution we started with the system…

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3x + 2y = 21 2x + y = 13

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We then multiplied both sides of the top equation by -3 and the bottom equation by 2 to obtain the equivalent system…

3x + 2y = 21 2x + y = 13

-6x + -4y = -42 -6x + -3y = -39

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We could continue in this manner by replacing the bottom equation in the

system above by the sum of the bottom equation and the top equation to obtain the

equivalent system of equations…

-6x + -4y = -42 -6x + -3y = -39

-y = - 3 -6x + -3y = -39

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…and if we then divide both sides of the top equation above by -3 we obtain the

equivalent system…

-y = - 3 -6x + -3y = -39

y = 3 2x + y = 13

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While both systems look quite different they are equivalent in the sense that they

possess the same solution set.

y = 3 2x + y = 13

That is, the ordered pair (x,y) is a solution of the first system if and only if it is also a solution of second system. However, the second system is easier to solve than the first system because the bottom equation tells us explicitly what number y has to be.

3x + 2y = 21 2x + y = 13next

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We can continue this process by replacing our system by an equivalent system in which the top equation contains only x as the variable. More specifically, we can multiply both sides of the bottom equation above by -1 to obtain the equivalent system…

y = 3 2x + y = 13

-y = -3 2x + y = 13

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Next we can replace the top equation in our system by the sum of the top and bottom equation

-y = -3 2x + y = 13

-y = -3

2x = 10

2x + 0 = 10

to obtain the equivalent system…

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Finally, we may divide both sides of the top equation in our system by 2

y = 3

x = 5

-y = -3

2x = 10

and the bottom equation by -1 to obtain the equivalent

system of equations…

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y = 3

x = 5

There is a tendency to think of the system

3x + 2y = 21 2x + y = 13

as being the solution of the system

In actuality, it is just an equivalent system of equations that happens to be “trivial”

to solve.

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It is similar to asking a question of the type, “What was the color of Paul Revere’s white horse?” Namely, it’s still a question

but one that already contains its own answer.

To continue the above analogy, suppose we were told that the color of

Paul Revere’s horse was the color of freshly fallen snow.

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This would implicitly tell us that in explicit This would implicitly tell us that in explicit terms the color of the horse was white.terms the color of the horse was white.

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In this context the system…

3x + 2y = 21 2x + y = 13

…tells us implicitly what the system…

tells us explicitly.

y = 3

x = 5

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The above discussion becomes more visual if we think in terms of the geometric

graphs of the two equations in the system…

3x + 2y = 21 2x + y = 13

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Namely, the geometric graph of the set consisting of all ordered pairs (x,y) for

which 2x + y = 13 is the line L1 (which is determined by the two intercepts (0,13) and

(6.5,0)).

In a similar way the graph of the set ofordered pairs (x,y) for which 3x + 2y = 21 is the line L2 (which is determined by the

two intercepts (0,10.5) and (7,0)).

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The graph of 2x + y = 13 is

(0,13)

(6.5,0)

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L1

When x = 0, y = 13

When y = 0, x = 6.5

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The graph of 3x – 2y = 21 is

(0,10.5)

(7,0)

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When x = 0, y = 10.5

When y = 0, x = 7

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L2


(0,13)

(6.5,0)

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(0,10.5)

(7,0)

L2

Clearly these two lines meet at one and only one point, and we have shown that

this point is (5,3).

(5,3)

2x + y =102x + y =10

2x + 3y =132x + 3y =13

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In our graphical solution, we already knew from the

algebraic solution that the point of intersection of the two lines was (5,3). However, notice from the graph that if we didn’t already know this fact, it would have been difficult to pinpoint exactly where the two lines meet.

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Note

For example, by looking at the graph, it is difficult to distinguish between such

points as (5,3) and (4.98,3.01).

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Thus, whenever possible, especially when an exact

answer is required, we should strive for an algebraic solution.

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Note

However, some problems are too complex to allow for an exact algebraic solution. In those cases, we use graphical methods to approximate the exact answer, and then

use trial-and-error to refine our approximation.

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The following systems…

each represents a pair of lineswhose point of intersection is (5,3).

-6x + -4y = -42 -6x + -3y = -39

-y = - 3 -6x + -3y = -39

y = 3 2x + y = 13

-y = -3 2x + y = 13

-y = -3

2x = 10

y = 3

x = 5

3x + 2y = 21 2x + y = 13


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In particular, the above system represents the

intersection of the vertical line x = 5 and

the horizontal line y = 3.

(5,3)

x = 5x = 5

y = 3y = 3

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y = 3

x = 5

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next A nice feature about mathematical

proofs is that there is usually only one correct answer to a question but never just

one correct way to find the answer. For example, we chose to eliminate x in the system…

…in order to first find the value of y. It would have been just as logical to first

eliminate y and instead find the value of x.

3x + 2y = 21 2x + y = 13

This would have led to a different set of equivalent systems, but each system would have as its solution set {(5,3)}.

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In fact we could have rewritten 2x + y = 13 in the form y = 13 – 2x and then replaced y

by 13 – 2x in the equation 3x + 2y = 21 to obtain the equation…

which is linear in x. 3x + 2(13 – 2x) = 21

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Solving this equation leads to the steps…

3x + 2(13 – 2x) = 21

3x + 26 – 4x = 21

3x– 4x + 26 = 21 -x = -5

x = 5

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Rewriting the linear equations in the form

y = mx + b, often referred to as the method of substitution, does not lend

itself very well to situations that deal with linear systems of equations that have

more than two unknowns.

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Note

In fact, even in the case of two unknowns, the arithmetic can become a bit “messy”. For example, 3x + 7y = 9

would become y = -3/7x + 9/7.

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next Systems of equations often arise when

we try to solve real world” problems algebraically.

For example, the system 3x + 2y = 21 2x + y = 13

2 boxes of chocolate chip cookies and 2 boxes of chocolate chip cookies and 1 box of mint chip cookies cost $13. 1 box of mint chip cookies cost $13. However, 3 boxes of chocolate chip However, 3 boxes of chocolate chip

cookies and 2 boxes of mint chip cookies cookies and 2 boxes of mint chip cookies cost $21. How much does each box of cost $21. How much does each box of

mint chip cookies cost?mint chip cookies cost?

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might arise if we were trying to solve the following problem…

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Sometimes using the generic letters x and y causes us to confuse which letter

represents the cost of a box of chocolate chip cookies and which letter represents

the cost of a box of mint chip cookies. In this context, it might be easier to keep track of the two items if we were to use, say, c to represent the cost of a box of chocolate chip cookies and m represent the cost of a box of mint chip cookies. In that way our system…

3x + 2y = 21 2x + y = 13

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3c + 2m = 21 2c + m = 13

becomes…

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When we solve this system and find that c = 5 and m = 3, we immediately know

that the cost of each box of mint chip cookies (that is: m) is $3.

As a final note, keep in mind that the “short cut” solution we gave earlier for

this exercise is the one that is most often used in solving systems of equations.

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Note

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However, it’s important to understand the sequence by which the implicit system is translated, step by step, into the explicit system (such as how we transformed…

Note

into 3x + 2y = 21 2x + y = 13

y = 3

x = 5

by a sequence of equivalent systems of equations).

key stone problem… key stone problem… next set 21 © 2007 herbert i. gross

Documents

value of x

gross slide

gross solution

gross notes

keystone problem

system of equations

equivalent system

linear system