Let's consider the scenario: "You have 2 times the number of tokens as your friend. After giving away 4 tokens, you have 16 tokens left. How many tokens did your friend start with?"

We can represent this statement in a few different ways. We can use pictures to represent the scenario.

A pictorial representation showing two pouches and four greyed-out tokens is equal to 16 colored tokens.

We can also use algebra tiles to represent the scenario. The algebra tiles would look like this:

Algebra tiles: Two positive variable tiles and four negative unit tiles is equal to 16 positive unit tiles

The equation that represents this scenario is: 2x-4=16

There are some key words to look out for when writing a sentence as an equation.

Addition	Subtraction	Multiplication	Division	Equal
plus	minus	times	divided by	is/are
the sum of	the difference of	the product of	the quotient of	equals
increased by	decreased by	multiplied by	separated into equal parts	amounts to
total	fewer than	of	split	totals
more than	less than	twice	equally shared
added to	subtracted from	groups/lots of

Examples

Example 1

Adam thought of a number, doubled it and added 5. The result was 13.

Write an equation that represents this scenario.

Worked Solution

Create a strategy

There is an unknown number so we should use a variable to represent it. 'Doubled' means multiplied by 2 and we are told he added 5.

Apply the idea

\displaystyle n \cdot 2 + 5	\displaystyle =	\displaystyle 13	Write the equation
\displaystyle 2n+5	\displaystyle =	\displaystyle 13	Simplify

Create a model using algebra tiles to represent this scenario.

Worked Solution

Apply the idea

Algebra tiles: Two positive variable 'n' tiles and five positive unit tiles is equal to 13 positive unit tiles.

Example 2

What scenario could the equation 3t=2t+12 be representing?

Worked Solution

Create a strategy

We should start by stating what t represents.

Apply the idea

Here is one possible scenario:

Let t be the number of tickets included in a season membership to an off Broadway theater.

3 season memberships have 12 more tickets than 2 season memberships.

Reflect and check

How many different scenarios can you come up with? What do all the scenarios have in common? What is different?

Example 3

Write the statements as equations, where x represents the number.

Four times a number decreased by \dfrac{3}{4} is equal to twice the number plus \dfrac{5}{6}.

Worked Solution

Create a strategy

Use a variable to represent the unknown number. 'Four times' means multiply by 4 while twice means multiply by 2. Decreased represents subtraction.

Apply the idea

4x-\dfrac{3}{4}=2x+\dfrac{5}{6}

The quotient of twenty and a number increased by \dfrac{11}{12} equals the product of three and a number minus \dfrac{1}{2}.

Worked Solution

Create a strategy

Use a variable to represent the unknown number. 'Quotient' means division and 'product' means multiplication. 'Increased' represents addition.

Apply the idea

\dfrac{20}{x}+\dfrac{11}{12}=3x-\dfrac{1}{2}

Idea summary

An equation is a mathematical sentence stating that two expressions are equal.

Equations often contain letters or symbols used to represent an unknown quantity. These are called variables. We look for keywords to help us determine which operations are being used in an equation.

Addition	Subtraction	Multiplication	Division	Equal
plus	minus	times	divided by	is/are
the sum of	the difference of	the product of	the quotient of	equals
increased by	decreased by	multiplied by	separated into equal parts	amounts to
total	fewer than	of	split	totals
more than	less than	twice	equally shared
added to	subtracted from	groups/lots of

Outcomes

7.PFA.3

The student will write and solve two-step linear equations in one variable, including problems in context, that require the solution of a two-step linear equation in one variable.

7.PFA.3a

Represent and solve two-step linear equations in one variable using a variety of concrete materials and pictorial representations.

7.PFA.3d

Write a two-step linear equation in one variable to represent a verbal situation, including those in context.

7.PFA.3e

Create a verbal situation in context given a two-step linear equation in one variable.

4.01 Write and represent two-step equations

Ideas

Write and represent two-step equations

Examples

Example 1

Example 2

Example 3

Outcomes

7.PFA.3

7.PFA.3a

7.PFA.3d

7.PFA.3e

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