Evaluate Expressions and Write Them from Word Problems | Grade 7 Math | LevelKite

You already know how to read an expression such as $3x + 5$ . The letter $x$ is a variable: it holds the place of a number. The expression has two terms, $3x$ and $5$ , which are the parts you add together. The coefficient $3$ is the number that multiplies the variable, and the constant $5$ is the number that stands alone.

Now look at what the expression does. It is a rule for working out a quantity: take a number, multiply it by $3$ , then add $5$ . Because $x$ can stand for many different numbers, one rule gives many results.

In this lesson you will use that rule in two directions. First you will evaluate an expression, which means you find its value when the variable is a given number. Then you will write an expression that matches a word problem.

To evaluate an expression, you substitute, which means you replace the variable with the given number. Substituting does not change the rule. It only swaps the letter for the number it stands for.

Every expression in this lesson is a set of terms joined by plus or minus signs. For an expression in that form, follow three steps:

Write the number in parentheses where the variable was. A number written next to a variable means multiply, and the parentheses keep that multiplication and any negative sign clear.
Multiply the number in front of each set of parentheses by the number inside. The result of a multiplication is a product.
Add or subtract from left to right.

You multiply before you add or subtract because that is the agreed order of operations.

Here is $3x + 5$ when $x = 4$ . Watch $3x$ become $3(4)$ , which is $12$ .

\text{When } x = 4:\quad 3x + 5 = 3(4) + 5 = 12 + 5 = 17

Worked example

Evaluate when the value is negative

Evaluate $10 - 3x$ when $x = -2$ .

Substitute. Write negative 2 in parentheses where x was. The rule stays the same: it still says to multiply by 3 and subtract the result from 10.
$10 - 3(-2)$
Multiply first, because 3x means 3 times x. A positive number times a negative number is negative, so 3 times negative 2 is negative 6.
$10 - 3(-2) = 10 - (-6)$
Rewrite the subtraction. Subtracting a number gives the same result as adding its opposite. The opposite of negative 6 is 6, the number at the same distance from zero on the other side.
$10 - (-6) = 10 + 6$
Add. No other operation is left, so the result is the value of the expression.
$10 + 6 = 16$

Answer

When $x = -2$ , the value of $10 - 3x$ is $16$ .

An expression with a variable, such as $10 - 3x$ , does not have just one value. Its value depends on the number you substitute. The table below evaluates $10 - 3x$ for four values of $x$ with the same three steps each time.

Look closely at two rows. When $x = 0$ , the product $3(0)$ is $0$ , so only the constant $10$ is left. When $x = 4$ , the product $12$ is greater than $10$ , so the value is negative.

One expression, four values of the variable

Value of x	Substitute	After multiplying	Value of the expression
$-2$	$10 - 3(-2)$	$10 - (-6)$	$16$
$0$	$10 - 3(0)$	$10 - 0$	$10$
$1.5$	$10 - 3(1.5)$	$10 - 4.5$	$5.5$
$4$	$10 - 3(4)$	$10 - 12$	$-2$

Key concept

An expression is a rule you can run and build

A linear expression tells you how to calculate a quantity from a variable.

Evaluating runs the rule. Substitute a number for the variable, then follow the order of operations. For the expressions in this lesson, that means multiply first, then add or subtract.
Writing builds the rule. Say what the variable counts and its unit, then choose the operations that match what happens in the situation.

The two skills work together. When you evaluate an expression you wrote for a value you can work out from the situation, you test it. If the values do not match, the expression needs fixing.

You have already turned short word phrases into expressions. A word problem gives you a whole story instead, about a quantity that depends on a number you do not know yet. Writing an expression turns that story into a rule. Work in this order:

Define the variable. Say exactly what it counts, its unit, and which numbers make sense for it.
Match each action in the story to an operation. Three actions appear in this lesson. When the same amount repeats for each unit of the variable, multiply that amount by the variable. When the story combines amounts, add them. When the story takes an amount away, subtract it.
Test your expression. Evaluate it for a small value that makes sense in the story, and compare the result with the story itself. A mismatch means the expression does not fit. A match is a good sign, not proof.

Some stories need one operation. If you have $p$ pages of homework and finish $4$ of them, $p - 4$ pages are left. If each bag holds $6$ oranges, then $b$ bags hold $6b$ oranges. Other stories need two operations, as the next example shows.

Worked example

Write an expression, test it, then use it

In a quiz game, you start with $50$ points and earn $15$ points for each correct answer. Write an expression for your score. Then find your score after $6$ correct answers.

Define the variable. The number of correct answers is what can change, so let c be the number of correct answers. It must be a whole number (0, 1, 2 and so on), because you cannot answer part of a question correctly. You measure the score in points.
$c = \text{number of correct answers}$
Match the repeating amount to multiplication. Each correct answer adds 15 points. Because c is a whole number, adding 15 a total of c times is the same as multiplying 15 by c.
$15 \cdot c = 15c$
Match the starting points to addition. The story combines the 50 starting points with the points you earn, so add them. The 50 does not change when c changes, so it is the constant. Adding in the other order, 15c + 50, gives the same value for every c.
$50 + 15c$
Test the expression with a small value. With 2 correct answers, the story gives 50 plus 15 plus 15, which is 80 points. The expression gives 80 as well, so it passes this test.
$50 + 15(2) = 50 + 30 = 80$
Answer the question. Write 6 in parentheses in place of c, multiply first, then add.
$50 + 15(6) = 50 + 90 = 140$

Answer

The expression $50 + 15c$ gives your score in points after $c$ correct answers. After $6$ correct answers, your score is $140$ points.

Worked example

A quantity that goes down

At 6 p.m. the temperature is $4^\circ\text{C}$ . It then drops steadily, $1.5^\circ\text{C}$ every hour, until midnight. Write an expression for the temperature $h$ hours after 6 p.m. Then find the temperature at midnight.

Define the variable. Let h be the number of hours after 6 p.m. The story runs from 6 p.m. to midnight, so h can be any number from 0 to 6, including part of an hour. The unit for the temperature is degrees Celsius.
$0 \le h \le 6$
Match the repeating drop to multiplication. The temperature drops 1.5 degrees every hour. Because the drop is steady, the total drop after h hours is 1.5 times h, even when h includes part of an hour.
$\text{total drop} = 1.5h$
Match the drop to subtraction. A drop takes degrees away, so subtract the total drop from the starting temperature, 4. The variable term is negative 1.5h, so its coefficient is negative 1.5, and the constant is 4.
$4 - 1.5h$
Test the expression with a small value. After 2 hours, the story gives 4 minus 1.5 minus 1.5, which is 1 degree. The expression gives 1 as well, so it passes this test.
$4 - 1.5(2) = 4 - 3 = 1$
Find h for midnight, then substitute. Midnight is 6 hours after 6 p.m., so h is 6, which is in the allowed range. Multiply first: 1.5 times 6 is 6 ones and 6 halves, or 6 plus 3, which is 9.
$4 - 1.5(6) = 4 - 9$
Subtract. Taking away 4 brings you to 0, and taking away the remaining 5 brings you to negative 5. A negative temperature means degrees below zero.
$4 - 9 = -5$

Answer

For $0 \le h \le 6$ , the expression $4 - 1.5h$ gives the temperature in degrees Celsius $h$ hours after 6 p.m. At midnight the temperature is $-5^\circ\text{C}$ , which is $5$ degrees below zero.

The big idea: an expression is a rule for a quantity. To evaluate it, substitute the given number in parentheses, follow the order of operations, and watch the signs. To write one from a word problem, define the variable and its unit, then match each action in the story to an operation: multiply for an amount that repeats, add for amounts the story combines, and subtract for an amount the story takes away. Then test what you wrote. Evaluate it for a value you can work out straight from the story. If the two values differ, fix the expression.