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Win Elements · Algebra 1 · Notes

Evaluating an Expression by Substitution

Skill 0.06 Standard: pre 6.EE.2c About 10 minutes

Substitution swaps a letter for a number. The parentheses you write while doing it are not decoration; they are what keeps the number intact.

You have already used substitution twice as a checking tool. In Skill 0.04 it exposed 3x + 2 pretending to be 5x, and in Skill 0.05 it exposed a distribution that stopped early. Now it becomes the task itself.

The work is mechanical: replace each variable with its given value, then evaluate using the order of operations from Skill 0.03. Almost every error happens in the replacing, not the evaluating.

1. Replace, then evaluate

Treat the variable as a slot. Copy the expression, drop the value into every slot, and only then start calculating. Keeping those two phases separate is what makes the work reliable.

3x + 7 3(−2) + 7 −6 + 7 1 put −2 in multiply add Replace first. Calculate second. Never both on the same line.
Figure 1. Three short lines beat one long one. If the answer is wrong, you can point to the line where it went wrong.

2. What the parentheses are for

A variable is a single slot holding a single number. When the value is negative, the minus sign is part of that number, not an operation floating nearby. Parentheses are how you keep the two ideas apart on paper.

2 2 this slot holds x (−3) Given: x = −3 The parentheses hold the minus sign and the 3 together as one number in the slot. One slot, one number, one pair of parentheses.
Figure 2. The exponent belongs to whatever fills the slot. Write the parentheses and the exponent automatically covers the whole value.
Key idea

Put parentheses around every substituted value, every time. It costs nothing when the value is positive and it saves the answer when the value is negative.

3. The mistake: losing the sign or the parentheses

Skill 0.03 showed that an exponent takes only the base directly beneath it. Substituting −3 into 2x² without parentheses produces 2 · −3², where the exponent grabs the 3 alone and leaves the minus sign outside. The answer flips from 18 to −18.

Watch out: the minus sign must go inside

Evaluate 2x² when x = −3, written both ways.

NO PARENTHESES 2 · −3² the exponent hits only the 3 2(−9) −18 WITH PARENTHESES 2(−3)² all of −3 is the base 2(9) 18 Two marks on the page, and a 36-point swing in the answer.

Sense check: after substituting, look at every negative value and ask whether it is fully wrapped. If a minus sign is sitting outside a parenthesis, it is about to be misread.

4. Substituting into a subtraction

The other half of the trap is a value that lands next to a subtraction sign. Writing 5 − −4 is confusing, and students often quietly drop one sign and compute 1. With parentheses the expression reads 5 − (−4), which Skill 0.01 already turned into 5 + 4.

5 − x 5 − (−4) 5 + 4 9 substitute add the opposite total Dropping one of those two minus signs gives 1 instead of 9.
Figure 3. Two minus signs in a row look like a typing error, but each one is doing a different job. The parentheses show which is which.

5. The same value, four expressions

One value can produce very different results depending on where it lands. These four all use x = −3, and the middle column is the only part worth slowing down for.

EXPRESSION AFTER SUBSTITUTING VALUE x² (−3)² 9 −x² −(−3)² −9 5 − x 5 − (−3) 8 2x 2(−3) −6
Figure 4. Rows one and two differ by a single minus sign that never entered the slot. That sign is outside the substitution, so it is applied last.

6. Worked examples

Example A

Evaluate 3x + 7 when x = −2.

Substitute first: 3(−2) + 7. One negative factor makes the product negative, so 3(−2) = −6. Then −6 + 7 is a move six left and seven right.

3x + 7 = 1 when x = −2. Sense check: the result should be small, since −6 and 7 nearly cancel.

Example B

Evaluate x² − 4x when x = −5.

Substitute into both slots: (−5)² − 4(−5). The square gives 25. The product 4(−5) is −20, and subtracting −20 adds 20.

x² − 4x = 45 when x = −5. Sense check: both terms end up positive, so the answer must exceed 25.

Example C

A January morning in Big Bear Lake reads 14 degrees Fahrenheit. Convert it using the formula C = 5(F − 32) ÷ 9.

Substitute F = 14: C = 5(14 − 32) ÷ 9. Grouping first gives 14 − 32 = −18. Then 5(−18) = −90, and −90 ÷ 9 = −10.

The temperature is −10 degrees Celsius. Sense check: 14 degrees Fahrenheit is well below freezing, so a negative Celsius reading is expected.

Words to know
Substitute
To replace a variable with a given number.
Evaluate
To carry out the operations until a single number remains.
Value
The number a variable is currently standing for.
Slot
The place a variable occupies, which one whole number fills.
Formula
An expression with a fixed meaning, such as a temperature conversion.

7. A look ahead

Coming next: Skill 0.07, plotting and reading ordered pairs in four quadrants

Skill 0.07 puts two numbers together as a location on a grid. The order matters there in a new way, and two traps wait. One is swapping x and y. The other is counting a coordinate from the wrong axis, which lands the point somewhere it was never meant to be.

Check your understanding
  1. Explain why 2(−3)² and 2 · −3² give different answers.
  2. Describe the two phases of substitution and why they stay on separate lines.
  3. Explain why x² and −x² differ when x = −3, even though both are squared.
  4. A classmate evaluates 5 − x at x = −4 and gets 1. Name the mistake.
  5. Explain why writing parentheses around a positive value costs you nothing.
What to carry away