How to Represent Constraints by Equations | Math

April 22, 2015 thetasctest

How to Represent Constraints by Equations

For the TASC Test Assessing Secondary Completion™ Mathematics subtest, you must be able to represent constraints by equations, and by systems of equations. You must also be able to interpret solutions as viable or non-viable options in a modeling context.

One of the most common examples of this, according to the standards and the high emphasis topics that the TASC test uses to design the Mathematics subtest, is being able to represent equations describing nutritional and cost constraints on combinations of different foods.

It is unlikely that you will see this exact problem on the TASC test, as it is a generalization. However, the problems may be similar, and the example is helpful in setting up a situation for us to understand what these mathematical terms are referring to. In the example, we know that the inequality is defined by nutritional standards and cost constraints. In a problem like this, you may be asked to choose which option has the highest nutritional value with the lowest cost constraint. In other words, find the healthiest food at the cheapest price.

One we think through what the terms mean, and how they relate to one another, we can work through an actual problem. For this example, we will demonstrate how to represent solutions and constraints to systems of linear equations by using a table.

Review

There are some math skills you may already be practicing that will be important for learning this new skill. First, you may already know that you can use a table to model a problem and have practiced using tables in other types of problems. Additionally, you may know that the solution to a system of equations is a set of values that makes both equations true. For example:

5x – 4y = -3

y= -3x + 5

In this example, you could use the given information to solve for both x and y. You would do this by placing the expression that y is equal to into the first problem, and solving for either x. You would then place the solution into the other equation to solve for y. In some math problems, you may need to solve for just one of the variables. In others, you may need to solve for both. It just depends on what the problem is asking you to do.

5x – 4(-3x + 5) = -3

5x + 12x – 20 = -3

17x – 20 = -3

20 + (17x - 20) = (-3) + 20

17x = 17

17/(17x) = (17)/17

x = 1

y = -3(1) + 5

y = -3 + 5

y = 2

To be a system, the solution to the system of equations must be true for both equations. If you did not solve correctly, then the system would not function correctly. Emily Raskin, an educator at LearnZillion, points out that when students are asked to solve systems of equations, many make the common mistake of solving for only one of the variables. Whenever you are solving a system of equations or you are representing constraints by equation, you must solve for both variables.

Solving for Constraints 

Raskin gives the example problem:

A magnetic construction set has 38 bars. Make a total of 7 regular polygons. The polygons can be either pentagons or hexagons. Use all of the bars.

To solve this, you must first know what pentagons and a hexagons are. A pentagon has five sides, meaning you would use 5 bars per pentagon. A hexagon has six sides, meaning you would use 8 bars per hexagon. Remember that you must make a total of 7 regular polygons.

Simple multiplication shows us that neither 7 pentagons nor 7 hexagons can solve the problem. To make 7 pentagons, we would need 35 bars and would have 3 bars leftover. The problem requires us to use all of the bars; therefore, this cannot be the solution. To make 7 hexagons, we would need 42 bars. However, we only have 38.

We must find the right combination of pentagons and hexagons to use all of the bars. Raskin suggests using a table to solve these constraints.

The table must show all the possible combinations:

Polygons Made

Pentagons

Hexagons

0

7

1

6

2

5

3

4

4

3

5

2

6

1

7

0

This table shows all possible combinations. For example, we could make 1 pentagon and 6 hexagons to meet the requirement of 7 polygons. We are not yet sure whether or not this meets the other constraint: using only 38 bars.

To determine how we can meet this second constraint, we can add onto our table:

Polygons Made

Pentagons

Hexagons

Total Bars Used

0 (= 0 bars)

7 (= 42 bars)

42

1 (= 5 bars)

6 (= 36 bars)

41

2 (= 10 bars)

5 (= 30 bars)

40

3 (= 15 bars)

4 (= 24 bars)

39

4 (= 20 bars)

3 (= 18 bars)

38

5 (= 25 bars)

2 (= 12 bars)

37

6 (= 30 bars)

1 (= 6 bars)

36

7 (= 35 bars)

0 (= 0 bars)

35

In our table, we see that the only solution that uses exactly 38 bars is the combination of 4 pentagons and 3 hexagons. We used multiplication to figure out how many bars each polygon would use, and addition to add the solution of pentagons and hexagons to test every possible combination.

In solving for constraints by equation, be careful to check that your solution satisfies both constraints. Check your answer against both aspects of the problem. Raskin notes that many students skip this last step – and it can cost you points in the long run. 

For a step-by-step video of this math tip, check out Raskin’s LearnZillion lesson. You can also find tutorials and videos to help you master solving for constraints at OnlineMathLearning.com. Or if you’re still struggling, leave a comment below. A TASC test team member can answer your questions and provide you with the support you need to prepare for the TASC test Mathematics subtest.

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