Main content

## 8th grade

### Course: 8th grade > Unit 3

Lesson 9: Linear models- Linear graphs word problems
- Modeling with tables, equations, and graphs
- Linear graphs word problem: cats
- Linear equations word problems: volcano
- Linear equations word problems: earnings
- Modeling with linear equations: snow
- Linear equations word problems: graphs
- Linear equations word problems
- Linear function example: spending money
- Linear models word problems
- Fitting a line to data

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# Modeling with tables, equations, and graphs

See how relationships between two variables like number of toppings and cost of pizza can be represented using a table, equation, or a graph.

Math is all about relationships. For example, how can we describe the relationship between a person's height and weight? Or how can we describe the relationship between how much money you make and how many hours you work?

**The three main ways to represent a relationship in math are using a table, a graph, or an equation.**In this article, we'll represent the same relationship with a table, graph, and equation to see how this works.

**Example relationship:**A pizza company sells a small pizza for dollar sign, 6 . Each topping costs dollar sign, 2.

## Representing with a table

We know that the cost of a pizza with 0 toppings is dollar sign, 6, the cost of a pizza with 1 topping is dollar sign, 2 more which is dollar sign, 8, and so on. Here's a table showing this:

Toppings on the pizza left parenthesis, x, right parenthesis | Total cost left parenthesis, y, right parenthesis |
---|---|

0 | dollar sign, 6 |

1 | dollar sign, 8 |

2 | dollar sign, 10 |

3 | dollar sign, 12 |

4 | dollar sign, 14 |

Of course, this table just shows the total cost for a few of the possible number of toppings. For example, there's no reason we couldn't have 7 toppings on the pizza. (Other than that it'd be gross!)

Let's see how this table makes sense for a small pizza with 4 toppings.

Here's the cost of just the pizza:

Here's the cost of the start color #11accd, 4, end color #11accd toppings:

start color #11accd, 4, end color #11accd toppings dot dollar sign, 2 per topping equals dollar sign, start color #e07d10, 8, end color #e07d10

This leads to the total cost of

dollar sign, start color #1fab54, 6, end color #1fab54, plus, dollar sign, start color #e07d10, 8, end color #e07d10, equals, dollar sign, 14.

## Representing with an equation

Let's write an equation for the total cost y of a pizza with x toppings.

Here's the cost of just the pizza:

Here's the cost of x toppings:

x toppings dot dollar sign, 2 per topping equals x, dot, 2, equals, start color #e07d10, 2, x, end color #e07d10

So here's the equation for the total cost y of a small pizza:

Let's see how this makes sense for a small pizza with 3 toppings:

x, equals, start color #11accd, 3, end color #11accd because there are start color #11accd, 3, end color #11accd toppings

The total cost is 6, plus, 2, left parenthesis, start color #11accd, 3, end color #11accd, right parenthesis, equals, 6, plus, 6, equals, dollar sign, 12

## Representing with a graph

We can create ordered pairs from the x and y values:

Toppings on the pizza left parenthesis, x, right parenthesis | Total cost left parenthesis, y, right parenthesis | Ordered pair left parenthesis, x, comma, y, right parenthesis |
---|---|---|

0 | dollar sign, 6 | left parenthesis, 0, comma, 6, right parenthesis |

1 | dollar sign, 8 | left parenthesis, 1, comma, 8, right parenthesis |

2 | dollar sign, 10 | left parenthesis, 2, comma, 10, right parenthesis |

3 | dollar sign, 12 | left parenthesis, 3, comma, 12, right parenthesis |

4 | dollar sign, 14 | left parenthesis, 4, comma, 14, right parenthesis |

We can use these ordered pairs to create a graph:

$$

Cool! Notice how the graph helps us easily see that the total cost of the small pizza increases as we add more toppings.

## We did it!

We represented the situation where a pizza company sells a small pizza for dollar sign, 6, and each topping costs dollar sign, 2 using a table, an equation, and a graph.

What's really cool is we used these three methods to represent the same relationship. The table allowed us to see exactly how much a pizza with different number of toppings costs, the equation gave us a way to find the cost of a pizza with any number of toppings, and the graph helped us visually see the relationship.

Now let's give you a chance to create a table, an equation, and a graph to represent a relationship.

## Give it a try!

An ice cream shop sells 2 scoops of ice cream for dollar sign, 3. Each additional scoop costs dollar sign, 1.

## Comparing the three different ways

We learned that the three main ways to represent a relationship is with a table, an equation, or a graph.

**What do you think are the advantages and disadvantages of each representation?**

**For example, why might someone use a graph instead of a table? Why might someone use an equation instead of a graph?**

*Feel free to discuss in the comments below!*

## Want to join the conversation?

- Hello! I hope you are having a great day. I think that the advantages are that they can show a lot of information that is easily understood. Even the Table in functions can be easy to use and practical and you will find a lot of solutions for just one equation. Equations are also easier to find with small numbers and they also show the relationship between the x-axis and the y-axis. The disadvantages of Equations are that with big numbers, the answer will be weird. A Disadvantage of using the Table is that when you use decimals, the Table won't work. The Disadvantage of using a graph is that you can probably have two unpredictable variables. For me, I prefer using the table more than the graph and the equation. I think it is easier for me because I can double-check my answer with each number in the table. Thanks!(30 votes)
- I would argue that a 7-topping pizza is, indeed, not "gross". For example, the Costco Food Court combination pizza (which was discontinued in 2020) had six toppings on top of cheese. It was considered 'good pizza', based on all the positive reviews it received, as well as the despair people shared when Costco stopped selling it.

Another example is the supreme pizza at Papa Johns. This pizza also contains 6 toppings, and yet people still buy and enjoy it. Not to mention other chains, such as Pizza Hut, allow you to put up to 7 toppings on your pizza.

It really comes down to personal preference, but needless to say, I personally think that just because your pizza has 7+ toppings, doesn't mean that it's "gross".(13 votes) - Is there an easier way to learn how to read and write the
*xy*equations?(9 votes)- Watch someone doing it (in the videos for exemple) then practice to do the same.(1 vote)

- I think the Graph is easier, these questions were so easy it was hard to figure it out, I thought it was gonna be hard.(9 votes)
- Graphs are pictures that show how data changes or relates to each other. They are good for showing the big picture of your data, like how it goes up or down over time, or how it compares to other data. Graphs can also help you find patterns or unusual points in your data.

Tables are lists that show the exact values of your data. They are good for looking up specific numbers or comparing two numbers. Tables can also show different kinds of information in one place, like names, dates, and amounts.

Equations are formulas that show how variables affect each other mathematically. They are good for finding out what happens when you change one variable or another. Equations can also tell you the exact relationship between variables, while graphs can only show you an estimate.(6 votes) - i think that the advantage is that if u know how to do works from a graph than it will be really easy to put it in the plotting graph and the disadvantage is like when you have to find a missing number, but if you get it wrong then thats when the bigger problem begins.(6 votes)
- How do I do the write an equation one?(2 votes)
- Just look for a patteren and you will get it! Good luck(6 votes)

- For any people having trouble with the ice cream scoop equation, I have a simple explanation.

Because the base cost of a cone with no scoops is one dollar, (0,1) The equation has a plus-one.( y = x + 1 )

It's just like the base pizza price equation which is 6$ for a pizza with no toppings, (0,6) And each topping costs two dollars. so you are multiplying the amount of toppings by 2, and then you add the base price which is six. Therefore, the equation is, ( y = 2x + 6 )(4 votes) - the making graph involves more data as compared to data needed to represent function as an equation.(3 votes)
- advantage,summarize a large dataset in visual form easily compare two or three data sets, disanvanges, equire additional written or verbal explanation(3 votes)