You have collected 2 boxes of donations. Each box contains the same number of canned goods.
Your parents also donate 5 cans.
You receive another donation of 3 boxes of cans, each holding the same number of cans as the original boxes.

How many cans need to be in each box to reach your goal?

We can draw a pictorial model and write an equation to represent this situation. It would look like this:

2 boxes plus 5 cans plus 3 boxes equals 20 cans.

We can also use algebra tiles to represent this equation. It would look like this:

2 positive x tiles, 5 positive 1 tiles, 3 positive x tiles equals 20 positive 1 tiles.

This scenario represents the equation 2x+5+3x=20. The solution is x=3. We can verify this solution by using the substitution property.

\displaystyle 2(3)+5+3(3)	\displaystyle =	\displaystyle 20
\displaystyle 6+5+9	\displaystyle =	\displaystyle 20
\displaystyle 20	\displaystyle =	\displaystyle 20

When representing verbal situations as equations, we often see key words or phrases that indicate operations on our variable representing the unknown quantity.

Any operation can be used in algebraic expressions - in fact, any combination of operations can be used. To be able to turn written expressions into algebra, we can look for certain keywords to indicate addition, subtraction, multiplication, or division.

Addition	Subtraction	Multiplication	Division
plus	minus	times	divided by
the sum of	the difference of	the product of	the quotient of
increased by	decreased by	multiplied by	half
total	fewer than	each/per	split
more than	less than	twice	equally shared
added to	subtracted from	double

Often the order we write an expression or equation in follows the order of the situation presented. The words than and from connected to an operation indicate a change in the order of terms around the operation symbol.

Just like for operations, there are key words that indicate equality or grouping.

Equal	Parentheses
is/are	quantity
equals	times the sum
amounts to	times the difference
totals

Examples

Example 1

Write each situation as an algebraic equation.

Four less than twice a number equals the same number increased by six

Worked Solution

Create a strategy

Represent the 'number' with a variable. 'Less' indicates subtraction. 'Twice' means multiply by 2, and 'increased' indicates addition.

Apply the idea

Choosing the variable x as our variable and the term equals separating our expressions, we can write the equation:2x-4=x+6

Reflect and check

An equation represents a balance between two expressions. On one side, we start with twice a number, reduce it by four, and on the other, we simply add six to the same number. This following scale helps visualize the equation.

A balance scale.Two x tiles and six -1 tiles on the left, one x tile and six 1 tiles on the right.

The product of six and the sum of a number and three is the difference of the number and twelve

Worked Solution

Create a strategy

Use a variable to represent the 'number'. 'Product' indicates multiplication, 'sum' means addition, and 'difference' means subtraction.

Apply the idea

Notice, we are finding the product of a sum. So we need to multiply the entire sum x+3 by 6. This will require us to use parentheses: 6(x+3).

Choosing the variable x as our variable and the term is separating our expressions, we write the equation:6(x+3)=x-12

Reflect and check

To better understand our equation 6\left(x+3\right)=x-12, let's visualize it using algebra tiles.

On one side of the equation, we have 6\left(x+3\right), which represents six groups of x+3. This visually shows the repeated addition that is caused by multiplication.

On the other side of the equation, we have x-12, represented by one x tile and twelve negative unit tiles.

Algebra tiles. Six groups of one x and three 1 on the left equals one x and twelve -1 on the right.

The quotient of four more than triple a number and two is nine

Worked Solution

Create a strategy

Use a variable to represent the 'number'. 'Quotient' indicates division, 'triple' means multiplied by 3, and 'more than' indicates addition.

Apply the idea

The word and acts as the separator between the numerator and denominator for the fraction representing the quotient. This tells us that 'four more than triple a number' is the numerator and 'two' is the denominator.

Using the word is to separate the expressions, we have the equation\dfrac{3x+4}{2}= 9

Reflect and check

We can visualize the equation using algebra tiles. On the left are three x tiles and four 1 tiles, half grayed-out to represent dividing by 2. This is equal to nine 1 tiles on the right.

Algebra tiles. On the Left are three x tiles and four 1 tiles- half of the tiles are grayed out equals 9 tiles on the right.

Example 2

Carrie is going shopping to buy school supplies. She plans to buy 6 notebooks, 2 packages of pens and 1 three-ring binder. The total cost will be \$ 30.

She remembers that the binder costs twice as much as a notebook, and a pack of pens costs the same as six notebooks.

Write an algebraic equation to represent the situation.

Worked Solution

Create a strategy

Let's assign a variable to represent the cost of one notebook. Since the binder costs twice as much as a notebook and a pack of pens costs the same as six notebooks, we can write the costs of the binder and pens in terms of the notebook's cost. Finally, we'll use the total amount Carrie can spend to set up our equation.

Apply the idea

We know the sum of everything Carrie buys is \$30, so:

6 \text{ notebooks} + 2 \text{ pens} + 1 \text{ binder} = \$30

Let n represent the cost of one notebook in dollars. So the cost of 6 notebooks is 6n.

The cost of one pack of pens is the same as the cost of 6 notebooks or 6n, so the cost of 2 packs of pens is 2 \cdot 6n.

The cost of one binder twice (or two times) the cost of a notebook, n, so the cost of a binder is 2n.

Adding these up and setting them equal to \$30 we get the equation: 6n + 2\cdot6n + 2n = 30

Draw a pictorial model to represent the equation you wrote in part (a).

Worked Solution

Create a strategy

To create a pictorial model of the equation, we will use visual representations for each component of the shopping list: notebooks, packs of pens, and the binder.

Apply the idea

Let's use a circle to represent a notebook, a square for a pack of pens, and a triangle for the binder.

Since the binder costs twice as much as a notebook, we'll place two circles inside the triangle to show this relationship. Similarly, a pack of pens, costing the same as six notebooks, will have six circles within a square.

Carrie plans to buy 6 notebooks (6 circles), 2 packs of pens (2 squares, each containing 6 circles), and 1 binder (1 triangle containing 2 circles). The total cost represented by these symbols must equal the total cost of \$30.

A pictorial model: 6 circles ,2 squares with 6 circles inside each, 1 triangle with 2 circles inside equals a 10 and 20 dollar bills.

Carrie solves the equations using the information she knows. Her value for n is 1.5. Verify and interpret the solution.

Worked Solution

Create a strategy

To verify Carrie's solution, we'll substitute her value for n into our the equation and see if the result matches the total cost of \$30 .

Apply the idea

Substituting x = 1.5 into the equation:

\begin{aligned} 6\left(1.5\right) + 2\cdot6\left(1.5\right) + 2\left(1.5\right)&=30\\\\ 9+18+3 &= 30 \\\\ 30 &=30 \end{aligned}

This shows that the value of 1.5 for x makes the equation true, so it is the solution.

Example 3

Write a real world scenario that could be represented by the equation 2x + 2\left(x+4\right) = 23.

Worked Solution

Create a strategy

To create a real-world scenario for the given equation, we'll think of situations where two quantities, both related to the same unit \left(x\right), add up to a total.

Apply the idea

Imagine Jamie is planning a garden border that has two parallel sides of equal length, x, and two additional sides that are each 4\text{ ft} longer than the shorter sides. Jamie wants to enclose the garden with a small fence.

The equation 2x + 2\left(x+4\right) = 23 represents the total length of fencing needed to surround the garden. Here, 2x is the combined length of the shorter parallel sides, and 2\left(x+4\right) is the combined length of the longer sides. The total length of the fencing required is 23\text{ ft}.

Idea summary

Key phrases for operations in equations:

Addition	Subtraction	Multiplication	Division
plus	minus	times	divided by
the sum of	the difference of	the product of	the quotient of
increased by	decreased by	multiplied by	half
total	fewer than	each/per	split
more than	less than	twice	equally shared
added to	subtracted from	double

Key phrases for equality and additional notation needed in an equation:

Equal	Parentheses
is/are	quantity
equals	times the sum
amounts to	times the difference
totals

Outcomes

8.PFA.4

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

8.PFA.4a

Represent and solve multistep linear equations in one variable with the variable on one or both sides of the equation (up to four steps) using a variety of concrete materials and pictorial representations.

8.PFA.4c

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

8.PFA.4d

Create a verbal situation in context given a multistep linear equation in one variable.

2.03 Write and represent multistep equations

Ideas

Write and represent multistep equations

Examples

Example 1

Example 2

Example 3

Outcomes

8.PFA.4

8.PFA.4a

8.PFA.4c

8.PFA.4d

8.PFA.4f

8.PFA.4g

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