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Evaluating Linear Expressions

Substitute values into expressions and formulas, follow the order of operations, and handle negatives safely.

Practice this skill 18 min Evaluate linear expressions Leer en español
Key idea

What you'll learn

To evaluate an expression means to find its value when you know what the variable is. If a plumber charges 45h + 60 dollars for h hours of work, evaluating the expression for h = 3.5 tells you the actual bill.

In this lesson you will learn to:

  • substitute a number for each variable, safely, using parentheses,
  • follow the order of operations so you get the one correct value,
  • handle negative numbers, fractions and decimals without sign errors,
  • evaluate expressions with two or more variables,
  • work backwards: find a missing value when you already know the result.

Why it matters on the GED: evaluating is everywhere. Many questions give you a formula — a cost rule, a temperature conversion, a formula from the formula sheet — and ask for a value. Answer choices are built from the most common slips: doing operations in the wrong order or losing a negative sign. Careful substitution protects you from all of them.

Words to know

  • Evaluate — find the value of an expression for given numbers (evaluate 2x + 1 for x = 4: 2(4) + 1 = 9).
  • Substitute — replace a variable with a number (put 4 in place of x).
  • Variable — a letter standing for a number (x, h, m).
  • Coefficient — the number multiplying a variable; 3x means 3 \cdot x, so 3x at x = 4 is 12, not 34.
  • Order of operations — the agreed order for doing math: parentheses, exponents, multiplication and division (left to right), addition and subtraction (left to right).
  • Opposite — the number with the other sign (-x is the opposite of x; if x = -6, then -x = 6).
  • Integer — a whole number or its negative (-3, 0, 8).
  • Fraction bar — acts as a grouping symbol: work out the whole top and the whole bottom before dividing.
  • Formula — an expression or equation that describes a real rule (F = 1.8C + 32 converts Celsius to Fahrenheit).

All Math words in the glossary

Substitute with parentheses, then follow the order of operations

Step 1 — Substitute in parentheses. Every time you replace a variable, put the number in parentheses. This one habit prevents most mistakes.

4x - 7 \text{ at } x = 5 \;\rightarrow\; 4(5) - 7
Plug-in diagram for 4x − 7 at x = 5: the value x = 5 is substituted in parentheses to give 4(5) − 7, then multiply to get 20 − 7, then subtract to get 13.4x − 7substitute in ( )4(5) − 7multiply: 4(5) = 2020 − 7subtract13x = 5

Step 2 — Follow the order of operations.

OrderWhatExample
1Parentheses and other grouping (fraction bars)(3 + 5) \cdot 2 = 8 \cdot 2
2Exponents2 \cdot 3^2 = 2 \cdot 9
3Multiplication and Division, left to right12 \div 3 \cdot 2 = 4 \cdot 2 = 8
4Addition and Subtraction, left to right10 - 4 + 1 = 7

Many people remember PEMDAS. Note that multiplication does not always come before division — they share a step and go left to right. The same is true for addition and subtraction.

Examples:

  • 4x - 7 at x = 5: 4(5) - 7 = 20 - 7 = 13.
  • 45h + 60 at h = 3.5: multiply first, 45(3.5) = 157.5, then add: 157.5 + 60 = 217.5.
  • \dfrac{3x - 2y}{4} at x = 6, y = -3: top first: 3(6) - 2(-3) = 18 + 6 = 24; then 24 \div 4 = 6.

Why the order matters: 45h + 60 with h = 3.5 is $217.50. If you add first, (45 + 60)(3.5) = 367.5 — a $150 mistake. The expression says multiply 45 by the hours first.

Order of operations for 45h + 60 at h = 3.5. Left, correct: 45(3.5) + 60 becomes 157.5 + 60 = 217.5 because multiplication comes first. Right, wrong: adding first gives (45 + 60)(3.5) = 105(3.5) = 367.5. The two answers differ by 150 dollars.45h + 60 at h = 3.545(3.5) + 60157.5 + 60217.5(45 + 60)(3.5)105(3.5)367.5Multiply first: correctAdd first: wrongThe two answers differ by $150.

Working with negative numbers

Negative values cause most errors. These rules are worth memorizing:

OperationRuleExample
positive × negativenegative3(-4) = -12
negative × negativepositive(-2)(-3) = 6
subtracting a negativeadd the positive7 - (-12) = 19
adding a negativesubtract9 + (-6) = 3
negative ÷ negativepositive(-9) \div (-3) = 3

The parentheses habit in action. Evaluate -2x + 9 at x = -3:

-2(-3) + 9 = 6 + 9 = 15

Without parentheses you might write -2 - 3 + 9 and get 4, which is wrong: -2x means -2 times x.

The opposite of a variable. If x = -6, then -x = -(-6) = 6. A minus sign in front of a variable does not mean the answer is negative.

Subtracting a product. Evaluate 7 - 3x at x = -4:

7 - 3(-4) = 7 - (-12) = 7 + 12 = 19
Number line from −2 to 20 showing 7 − (−12): start at 7 and, because subtracting a negative means adding, move 12 units to the right to land on 19. So 7 − 3(−4) = 7 − (−12) = 19.7 − 3(−4) = 7 − (−12) = 7 + 12 = 19−202468101214161820start 719subtract −12 = move 12 right

Do the multiplication 3(-4) = -12 before the subtraction, then subtract the negative by adding.

Several variables, decimals, and working backwards

More than one variable. Substitute each value into its own spot. Write the expression first, then the numbers under it.

3a + 2b at a = 4, b = -5: 3(4) + 2(-5) = 12 + (-10) = 2.

Plug-in diagram with two variables: 3a + 2b with a = 4 and b = −5. Substituting gives 3(4) + 2(−5), multiplying gives 12 + (−10), and adding gives 2.3a + 2bsubstitute3(4) + 2(−5)multiply12 + (−10)add2a = 4b = −5

Decimals and money. Real-world formulas often use decimals. A truck rental costs 29.95 + 0.79m dollars for m miles. For 120 miles: 0.79(120) = 94.80, then 29.95 + 94.80 = 124.75, so the cost is $124.75.

Fractions. Treat the fraction bar like parentheses. For \dfrac{x + 9}{x - 1} at x = 3: top = 12, bottom = 2, value = 6.

Formulas. Many GED questions give a formula and ask for one value. Example: the temperature formula F = 1.8C + 32 at C = -15:

F = 1.8(-15) + 32 = -27 + 32 = 5

So -15°C is 5°F.

Working backwards. Sometimes you know the result and need the input. Undo the operations in reverse order.

The rental cost was $73.80 at 45 + 0.30m dollars. Subtract the fixed part first: 73.80 - 45 = 28.80. Then divide by the rate: 28.80 \div 0.30 = 96 miles.

You can also be asked for a missing constant. If 3x - k = 20 when x = -4, substitute: 3(-4) - k = 20, so -12 - k = 20, -k = 32, and k = -32.

Formula

The rules you use (memorize — not on the formula sheet)

Order of operations:

\text{Parentheses} \;\rightarrow\; \text{Exponents} \;\rightarrow\; \text{Multiply/Divide (left to right)} \;\rightarrow\; \text{Add/Subtract (left to right)}

Sign rules:

(-a)(-b) = ab \qquad (-a)(b) = -ab \qquad a - (-b) = a + b

Here a and b stand for any positive numbers.

The order of operations and the sign rules are not on the GED formula sheet; you must know them. However, many formulas you will evaluate are on the sheet — for example simple interest I = prt, perimeter and area formulas, and the slope formula. Evaluating them uses exactly the skills in this lesson.

Worked example

Example 1 (easy): One variable

Evaluate 6x - 11 when x = 4.

  1. Substitute in parentheses: 6(4) - 11.
  2. Multiply: 6(4) = 24.
  3. Subtract: 24 - 11 = 13.

The figure sums it up:

Plug-in diagram for 6x − 11 at x = 4: substitute to get 6(4) − 11, multiply to get 24 − 11, subtract to get 13. A note warns that 6x means 6 times x, not the number 64.6x − 11substitute in ( )6(4) − 11multiply: 6(4) = 2424 − 11subtract13x = 46x means 6 × xnot 64

Answer: 13

Check: 6x means "6 times x", not "64". 6 \times 4 = 24 ✓, and 24 - 11 = 13 ✓

Worked example

Example 2: Two variables with negatives

Evaluate -3x + 2y when x = -2 and y = -5.

  1. Substitute: -3(-2) + 2(-5).
  2. Multiply: -3(-2) = 6 (negative × negative) and 2(-5) = -10.
  3. Add: 6 + (-10) = -4.

The figure sums it up:

Number line from −6 to 8 for −3x + 2y at x = −2, y = −5. A blue arrow from 0 to 6 shows −3(−2) = +6. A red arrow from 6 moving 10 units left shows adding 2(−5) = −10, ending at −4.−3(−2) + 2(−5) = 6 + (−10) = −4−6−5−4−3−2−1012345678−3(−2) = +6+ 2(−5) = −10: move 10 leftend −4

Answer: -4

Check: do it on the calculator with the (−) key: (−) 3 × (−) 2 + 2 × (−) 5 enter → -4 ✓

Worked example

Example 3 (word problem): An electrician's bill

An electrician charges a $40 trip fee plus $65 per hour. The cost in dollars for h hours is 65h + 40. What does a 2.5-hour job cost?

  1. Substitute: 65(2.5) + 40.
  2. Multiply first: 65(2.5) = 162.5.
  3. Add the fee: 162.5 + 40 = 202.5.

The figure sums it up:

Bar model of the electrician bill 65h + 40 at h = 2.5: bars for hour 1 (65 dollars), hour 2 (65 dollars) and half an hour (32.50 dollars) make labor 65(2.5) = 162.50, dollars then the 40 dollars trip fee is added. Total: 202.50. dollars65h + 40 at h = 2.565hour 165hour 232.50½ hour40trip feelabor: 65(2.5) = 162.50+ 40Total bill: 162.50 + 40 = $202.50

Answer: $202.50

Check: 2 hours cost $130 and half an hour costs $32.50, so labor is $162.50; plus $40 is $202.50 ✓

Worked example

Example 4 (word problem): Simple interest from the formula sheet

Simple interest is I = prt (on the GED formula sheet), where p is the principal, r is the yearly rate as a decimal, and t is the time in years. Find the interest on $2,400 at 3.5% for 3 years.

  1. Convert the rate: 3.5\% = 0.035.
  2. Substitute: I = (2400)(0.035)(3).
  3. Multiply: 2400 \times 0.035 = 84, then 84 \times 3 = 252.

The figure sums it up:

Bar model for simple interest I = prt with p = 2400, r = 0.035 and t = 3: each year earns 2400 × 0.035 = 84, dollars shown as three equal bars labelled year 1, year 2 and year 3. Together they make 84 × 3 = 252. dollarsI = prt = (2400)(0.035)(3)one year: 2400 × 0.035 = $84$84year 1$84year 2$84year 33 years: 84 × 3 = $252 interest

Answer: $252

Check: 1% of $2,400 is $24, so 3.5% is 3.5 \times 24 = 84 dollars per year; for 3 years, 84 \times 3 = 252 ✓

Worked example

Example 5 (GED-hard): Work backwards

A car rental costs 45 + 0.30m dollars, where m is the number of miles driven. Dana's bill was $73.80. How many miles did she drive?

  1. Set the expression equal to the bill: 45 + 0.30m = 73.80.
  2. Undo the addition: 0.30m = 73.80 - 45 = 28.80.
  3. Undo the multiplication: m = 28.80 \div 0.30 = 96.

The figure sums it up:

Working backwards for the rental 45 + 0.30m = 73.80. Forward row: m miles, times 0.30, plus 45, gives 73.80. Backward row, read right to left: 73.80 minus 45 is 28.80, divided by 0.30 is 96 miles.Forward: how the bill is builtm miles× 0.300.30m+ 4573.80Backward: undo in reverse order96 miles÷ 0.3028.80− 4573.80

Answer: 96 miles

Check: substitute m = 96: 45 + 0.30(96) = 45 + 28.80 = 73.80 ✓

Calculator tip

Calculator tip (TI-30XS MultiView)

The calculator is very helpful for evaluating expressions with decimals or large numbers. Three habits make it reliable:

  1. Use the (−) key for negative numbers, not the subtraction key −. To enter -3, press (−) 3.
  2. Type parentheses around every substituted negative. For -2x + 9 at x = -3, type (−) 2 × ( (−) 3 ) + 9 enter → 15.
  3. Group the top and bottom of a fraction. For \dfrac{3x - 2y}{4}, either use the n/d key and type the whole numerator in the top box, or type ( 3 × 6 − 2 × ( (−) 3 ) ) ÷ 4. Without the outer parentheses, the calculator divides only the last term by 4.

The TI-30XS follows the order of operations automatically, so you can type the expression just as it is written. When the answer is a fraction, press toggle to switch between fraction and decimal form.

When not to use it: simple integer substitutions like 4x - 7 at x = 5 are faster in your head, and the GED puts some arithmetic in a section without a calculator.

Common mistake

Common mistakes

  • Writing the number next to the coefficient. Wrong: 4x at x = 5 is 45. Right: 4x = 4 \cdot 5 = 20.
  • Adding before multiplying. Wrong: 45h + 60 at h = 3.5 is (45 + 60)(3.5) = 367.5. Right: multiply first, 157.5 + 60 = 217.5.
  • Negative times negative. Wrong: -2(-3) = -6. Right: -2(-3) = +6.
  • Subtracting a negative. Wrong: 7 - 3(-4) = 7 - 12 = -5. Right: 3(-4) = -12 and 7 - (-12) = 19.
  • Dividing only part of a fraction. Wrong: \dfrac{18 + 6}{4} = 18 + 1.5. Right: add the top first: \dfrac{24}{4} = 6.
  • Forgetting the fixed part in "work backwards" problems. Wrong: 73.80 \div 0.30 = 246 miles. Right: subtract the $45 first, then divide.
Test strategy

GED test strategy

  • Write the expression, then substitute in parentheses. This is worth the extra five seconds on every question.
  • Estimate first. For 65h + 40 at h = 2.5, a rough answer is 65 \times 2.5 \approx 160, plus 40, about $200. That rules out choices like $102.50 or $262.50 at once.
  • Spot the trap answers. If you see one choice that is your answer with the opposite sign, slow down and recheck your negatives.
  • Work-backwards questions: you can also plug each answer choice into the expression and see which one gives the stated result.
  • Check the units. Is the answer in dollars, miles, minutes or degrees? Questions sometimes ask for the hours, not the cost.
  • Read twice. Make sure you substituted each value into the right variable, especially when there are two or three variables.

Check yourself

Evaluate linear expressionsNo calculator

What is the value of 4x - 7 when x = 5?

Answer choices
Hint

4x means 4 times x. Multiply before you subtract.

Check yourself

Evaluate linear expressionsNo calculatorFill in

Evaluate 3a + 2b when a = 4 and b = -5.

Type your answer in the box. Fractions like 3/4 are fine.

Hint

Substitute both values in parentheses. What is 2 \times (-5)?

Summary

  • Evaluate = substitute the given numbers and compute.
  • Put every substituted number in parentheses, especially negatives.
  • Follow the order of operations: parentheses, exponents, multiply/divide, add/subtract.
  • Sign rules: negative × negative = positive; subtracting a negative = adding.
  • A fraction bar groups the whole top and the whole bottom.
  • To work backwards, undo the operations in reverse order, or plug in the answer choices.
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