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Rearranging an Equation in Three Variables

1.9.1 Rearranging an Equation in Three Variables

1.9.1 • Rearranging an Equation in Three Variables

Warm Up

In an earlier lesson, you saw the equation V+F−2=EV+F−2=E, which relates the number of vertices, faces, and edges in a Platonic solid.

In questions 1 – 2, write an equation that makes it easier to find the number of vertices in each of the Platonic solids described:

1.

An octahedron (shown here), which has 8 faces

A line drawing of an octahedron, a geometric shape with eight triangular faces. Solid lines show visible edges, and dotted lines represent hidden edges, illustrating its three-dimensional structure.

Compare your answer: V=E−6V=E−6

V+8−2=EV+8−2=E

V+6=EV+6=E

V=E−6V=E−6

2.

An icosahedron, which has 30 edges

Compare your answer: V=32−FV=32−F

V+F−2=30V+F−2=30

V+F−2+2=30+2V+F−2+2=30+2

V+F=32V+F=32

F=32−FF=32−F

3.

A Buckminsterfullerene (also called a “Buckyball”) is a polyhedron with 60 vertices. It is not a Platonic solid, but the numbers of faces, edges, and vertices are related the same way as those in a Platonic solid.

Write an equation that makes it easier to find the number of faces a Buckyball has if we know how many edges it has.

Compare your answer: F=E−58F=E−58

V+F−2=EV+F−2=E

60+F−2=E60+F−2=E

58+F=E58+F=E

58−58+F=E−5858−58+F=E−58

F=E−58F=E−58

Why Should I Care?

Illustration of a woman holding a piggy bank, surrounded by icons representing savings goals such as a car, house, calendar, credit card, calculator, airplane, graph, and documents with money symbols.

Alex's brother can use algebra to budget his paycheck from his job at a pizza place. He uses some money for food, some for going out with his friends, and some, he saves for college.

If he uses variables to represent the different ways he uses his money, he can put those variables into an equation to make sure he has enough money to last until his next paycheck.