Writing Algebraic Expressions

Turning Sentences into Equations

In Math class, you will often need to write equations or expressions based on a scenario to perform some math and solve the question. You will want to express unknown terms using variables.

Look for terms like "a number" which can be represented with a variable like $x$. Then you will need to add operators ($+, -, *, /$). Look for terms like "increase", "decrease", "double", "triple", "half". Finally, if you need to write an equation ($=$) look for terms like "is" or "equals".

Outlined below is a table that summarizes these concepts:

Part of Equation	Example	Keywords
Variables	$x, y$	"A number"
Operators	$+ - * /$	"increase", "decrease", "double", "triple", "half", "more", "less"
Equal Sign	$=$	"is", "equals"

If we had to write an expression for the phrase "double a number is $14$", the number would be represented by $x$, we need to multiply it by $2$ since it says "double" and the term is equal to 14 because it says "is":

$2x = 14$

Write an expression for the phrase 'a number decreased by $6$ is $5$'.

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To write this as an expression we break down the phrase into several parts:

There is an unknown number. We will label it $x$.
It is decreased by $6$. We will subtract $6$.
It is $5$. We will make the expression equal to $5$.

Therefore, we can write the final equation as such:

$\boldsymbol{x - 6 = 5}$

Writing Equations for Word Problems

Writing equations is very important for solving word problems! Use what we have learned to solve the question below. Remember, each unknown is represented with a variable.

Sometimes there are several unknowns. Use separate variables to represent each one. It is also good practice to write out what each variable represents.

Example

Elmer earns $$5$ per hour when they babysits $1$ child. They earn $$8$ per hour when they babysits $4$ children. Write an expression to represent their earnings.

First, we can outline what each of the variables represents:

Let $E$ represent Elmer's earnings in dollars ($)
Let $x$ represent the number of hours they babysat $1$ child
Let $y$ represent the number of hours they babysat $4$ children

Elmer's earnings is the sum of the money earned from babysitting $1$ child and $4$ children. The earnings from babysitting $1$ child is $5x$ because they earn $$5/\text{hr}$ and $x$ represents the number of hours worked. Similarly, the earnings from babysitting $4$ children is $8y$ because they earn $$8/\text{hr}$ and $y$ is the number of hours worked.

Therefore, we can algebraically represent Elmer's total earnings as such:

$\boldsymbol{E = 5x + 8y}$

The length of a rectangle is 2 times the width of the rectangle. Let $x$ represent the width of the rectangle.

Write an expression to represent the length of the rectangle.

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We can represent the length of the rectangle as such:

$\boldsymbol{l = 2x}$

Write a simplified expression for the perimeter of the rectangle.

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The perimeter is $2$ times the sum of length and width. We can represent this as:

$P = 2 (l + x)$

Given that we can represent the length as $l=2x$, we can substitute this value into the equation:

$P = 2(2x + x)$

Finally, we can simplify this equation:

$P = 2(3x)$

$P = 6x$

Therefore, we can represent the simplified expression for a rectangle's perimeter as $\boldsymbol{P = 6x}$.

Suppose the width is $6 \; [\text{cm}]$. Find the perimeter of the rectangle.

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In order to determine the rectangle's perimeter, we can substitute $6$ for $x$ and solve:

$P = 6x$

$P = 6 (6)$

$P = 36 \; [\text{cm}]$

Therefore, we can determine that rectangles perimeter is $\boldsymbol{36 \; [\textbf{cm}^2]}$ when the width is $6 \; [\text{cm}]$.

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Polynomial Operations