# Let's Get Started

**TEKS Standards and Student Expectations**

**A(2)** Linear functions, equations, and inequalities. The student applies the mathematical process standards when using properties of linear functions to write and represent in multiple ways, with and without technology, linear equations, inequalities, and systems of equations. The student is expected to:

**A(2)(C)** write linear equations in two variables given a table of values, a graph, and a verbal description

**A(2)(I) **write systems of two linear equations given a table of values, a graph, and a verbal description

**Resource Objective(s)**

Create a verbal problem from a symbolic representation of a system of equations.

**Essential Questions**

What is a system of equations?

What words/phrases represent the different operation symbols?

**Vocabulary**

- System of Equations
- Sum
- Difference
- Quotient
- Product

# Systems of Equations

What is a system of equations? A **system of equations **is a set of two or more equations with the same variables.

We're going to investigate writing and solving systems of equations.

Below is an example of a two equation, two variable system.

$\left\{\begin{array}{l}t+m=2.10\\ 3t+3m=5.15\end{array}\right.$

While you can solve a problem like the one above by using graphs, tables, and algebraic methods, we're first going to concentrate on translating words into a system of equations.

# Writing a First Equation

To write systems of equations, you will need a few new tools for your math toolbox. Let’s take a look at a few examples.

**Example**

Akil caught 20 trout and bass while on a fishing trip. The total weight of his catch was 112 pounds. The average weight of a trout was 3.5 pounds, and the average weight of a bass was 7 pounds. Which system of equations can be used to find *t*, the number of trout, and *b*, the number of bass, that Akil caught?

The first sentence will lead to one of the equations: *Akil caught 20 trout and bass.*

However, you have to understand that Akil caught 20 fish, some of which were trout, and the rest were bass. If you misread the problem as 20 of each kind of fish, you would not find the correct answer here.

What are some possibilities if Akil caught 20 fish and there were some trout and the rest were bass?

Think of four possibilities. Copy and complete the first four rows of the table into your notes.

Number of Trout | Number of Bass | Process | Total Number of Fish |

_______ | _______ | _______ | _______ |

_______ | _______ | _______ | _______ |

_______ | _______ | _______ | _______ |

_______ | _______ | _______ | _______ |

# Writing a Second Equation

Let's hope you'd never rely on getting your answer with one equation. We need to make sure the second equation is correct. Look at the problem again.

**Example 1:**

Akil caught 20 trout and bass while on a fishing trip. The total weight of his catch was 112 pounds. The average weight of a trout was 3.5 pounds, and the average weight of a bass was 7 pounds. Which system of equations can be used to find *t*, the number of trout, and *b*, the number of bass, that Akil caught?

**A**. *t* = 20 + *b* **C**. *t* + *b* = 112

3.5*t* + 7*b* = 112 3.5*t *+ 7*b* = 20

**B**. *t* + *b* = 20 **D**. *t* = 112 + *b*

3.5*t* + 7*b* = 112 3.5*t* + 7*b* =20

The second equation is going to come from sentences two and three.

*The total weight of his catch was 112 pounds. The average weight of a trout was 3.5 pounds, and the average weight of a bass was 7 pounds.*

You're looking for the number of trout and the number of bass that would weigh 112 pounds. This won't be easy to find, but play with your calculator until you come up with 112 pounds.

Here are a few wrong guesses I made. Don't feel bad if you can't find any that work right away.

You are trying to get to 112 pounds of fish. A first guess of 5 trout at 3.5 pounds each and 10 bass at 7 pounds each was too low. You raised the number of bass to 12, but it was still too low, only 101.5 pounds. Then a third guess of 5 trout and 14 bass was too high. Can you find four combinations of trout and bass to make 112 pounds? Don't take more than 5 minutes searching, but try to find at least two.

Copy and complete the first four rows of the table using your notes as you find combinations that work.

On the last row, write *t *and *b* and try to find the process for a total weight of 112.

Number of Trout | Number of Bass | Process | Total Number of Fish |

____________ | ____________ | ____________ | ____________ |

____________ | ____________ | ____________ | ____________ |

____________ | ____________ | ____________ | ____________ |

t |
b |
____________ | ____________ |

# Writing Equations without Tables

We got our answer, but it took a long time to make the tables and come up with an equation. You want to eventually write the equations without making the tables. Look at the problem again.

**Example 1**

Akil caught 20 trout and bass while on a fishing trip. The total weight of his catch was 112 pounds. The average weight of a trout was 3.5 pounds, and the average weight of a bass was 7 pounds. Which system of equations can be used to find t, the number of trout, and b, the number of bass, that Akil caught?

**A**. t = 20 + b ** C**. t + b = 112

3.5t + 7b = 112 3.5t + 7b = 20

**B**. t + b = 20 ** D**. t = 112 + b

3.5t + 7b = 112 3.5t + 7b =20

We know the answer is **B**.

*Akil caught 20 trout and bass while on a fishing trip.*

*The total weight of his catch was 112 pounds. The average weight of a trout was 3.5 pounds, and the average weight of a bass was 7 pounds*.

These sentences give the equation 3.5t + 7b = 112. Do you see the pattern? Trout are 3.5 pounds each, which gives 3.5t. Bass are 7 pounds each, which gives 7b. Again, the total weight is 112 pound,s so add 3.5t and 7b to get 112.

How many trout and bass were there?

In this resource, you don’t have to solve the system of equations, but can you tell from this table which combination of trout and bass is correct? Why?

Number of Trout | Number of Bass | Process | Total Number of Fish |

4 | 14 | 3.5(4)+7(14) | 112 |

8 | 12 | 3.5(8)+7(12) | 112 |

10 | 11 | 3.5(10)+7(11) | 112 |

20 | 6 | 3.5(20)+7(6) | 112 |

t | b | 3.5(t)+7(b) | 112 |

# Geometry Practice

Let’s try an example from Geometry.

The length of a rectangular table cloth is equal to triple the width. Which system of equations can be used to find the dimensions of the table cloth if the perimeter is 75 centimeters?

A. |
l = w + 32( l + w) = 75 |
C. |
l = 3w2( l + w) = 75 |

B. |
l = 3w2 l + 6w = 75 |
D. |
l = w + 32 l + 6w = 75 |

# Vocabulary Activity

# Journal Activity—Writing Your Own Problems

Now, you are going to create a problem for a system of equations.

Eight systems of equations are given in the following table. Choose one system of equations, then write statements in the box to the right of the system of equations that would translate from a verbal description to the given system of equations.

Read the two examples that are given below.

**Example 1**

$\left\{\begin{array}{l}x+1.5y=4\\ x=y-1\end{array}\right.$

Mary spent \($\)4.00 on cheeseburgers and french fries. Cheeseburgers cost \($\)1.00, and french fries cost \($\)1.50. The number of cheeseburgers she bought is one less than the number of french fries.

**Example 2:**

$\left\{\begin{array}{l}x+y=5\\ 4x+3y=2\end{array}\right.$

Marcus makes a profit of \($\)5.00 when he makes and sells one of his T-shirts. This week he made four shirts and sold three shirts, for a profit of \($\)2.00.