Algebra 1
Expressions vs. equations; evaluating expressions
Algebra has two main characters, and a lot of early confusion comes from mixing them up. One is an expression, a phrase that names a value, like 3x + 2. The other is an equation, a full sentence making a claim, like 3x + 2 = 14.
You evaluate an expression; you solve an equation. Trying to "solve" an expression, or "evaluate" an equation, is like trying to answer a noun, and it derails what comes next.
To tell them apart, look for an equals sign. An expression is a noun phrase, like "three more than twice a number," with no verb and no =. An equation is a full sentence, with the = acting as the verb that claims a balance. No equals sign, it's an expression; equals sign present, it's an equation.
Evaluating an expression is where the mystery box pays off. Picture 3x + 2 as a row of seats reserved for a guest who hasn't arrived; x is the empty chair. "Evaluate at x = 4" is the guest sitting down: every x becomes a 4. So 3x + 2 at x = 4 is 3(4) + 2.
Now just follow order of operations from Lesson 1.3: 3(4) is 12, then 12 + 2 is 14. Write the steps out, 3(4) + 2 = 12 + 2 = 14, rather than jumping straight to 14, because the careful-substitution habit is what keeps signs and order from slipping.
Drag x, and watch the expression compute:
3(0) + 2 = 2
And signs will show up, so here's one with a negative. Evaluate 5 − 2x at x = −1. Drop −1 into the seat, keeping it in parentheses so the signs stay visible: 5 − 2(−1). That's 5 − (−2), and subtracting a negative adds (straight from Lesson 1.5), so 5 + 2 = 7.
The parentheses around the −1 are what save you here; without them it's easy to lose a sign.
One last thing to notice, not yet to formalize: structure matters. 2(x + 3) is not the same as 2x + 3, because the 2 multiplies everything inside the parentheses, not just the x. You can see it with a number. At x = 3, 2(3 + 3) = 12 but 2(3) + 3 = 9. There's a proper rule behind this that comes later in the course; for now, just respect the parentheses.
The input-to-output idea from Lesson 1.1 is hiding here too. Evaluating 3x + 2 at different values is feeding inputs through a machine: in goes 4, out comes 14. That machine gets a name later in the course; for now, just see that the same input-to-output pattern is here.
New words
- 1.6.d1 Expression: a mathematical phrase with no equals sign, e.g. 3x + 2. It names a value once you know the variable; you evaluate it.
- 1.6.d2 Equation: a full sentence with an =, e.g. 3x + 2 = 14. It claims two things are equal; you solve it.
- 1.6.d3 Evaluate / substitute: replace the variable with a given number and compute (the reserved seat finally has its guest sit down).
Worked example
- 1.6.w1 Expression or equation? 3x + 2. No equals sign, so it's an expression, a phrase that names a value once you know x.
- 1.6.w2 Expression or equation? 3x + 2 = 14. It has an =, so it's an equation, a sentence claiming the two sides match.
- 1.6.w3 Evaluate 3x + 2 at x = 4. The seat holds 4, so 3(4) + 2 = 12 + 2 = 14. Each step written out, no jumping.
- 1.6.w4 Evaluate 5 − 2x at x = −1. Keep the −1 in parentheses: 5 − 2(−1) = 5 + 2 = 7. Subtracting a negative adds, so watch that sign.
- 1.6.w5 Structure check: evaluate 2(x + 3) and 2x + 3 at x = 3. The first is 2(3 + 3) = 2(6) = 12; the second is 2(3) + 3 = 9. Different values, so they're different expressions; the parentheses change the meaning.
For one that just lets the method click, evaluate 2x + 1 at x = 5. The seat holds 5, so 2(5) + 1 = 10 + 1 = 11. Substitute, then compute in order, nothing more.
There's nothing to "solve" in an expression, only a value to find once you know the variable. And whenever you substitute a negative, wrap it in parentheses (5 − 2(−1), never 5 − 2−1); that one habit prevents most dropped-sign slips.
Check yourself
- 1.6.c1 Is 7y an expression or an equation? How do you know, and what would you do with it? (An expression, because it has no equals sign, so you'd evaluate it once you know y, not solve it.)
- 1.6.c2 Evaluate 4x − 1 at x = 2, then at x = −2. What changed? (At x = 2: 4(2) − 1 = 7. At x = −2: 4(−2) − 1 = −8 − 1 = −9. Switching the input's sign flipped the first term from +8 to −8.)
- 1.6.c3 Does 2(x + 3) equal 2x + 3? Test it with x = 5. (No: 2(5 + 3) = 16 but 2(5) + 3 = 13, so they're different; the 2 has to reach the 3 inside as well.)
You can now tell an expression from an equation by spotting the equals sign, evaluate an expression by substituting and then computing in order, keep negatives safe inside parentheses, and see why the placement of a minus sign changes −2² and (−2)².
Deciding for each one whether to identify it or evaluate it is the work a page of all-the-same never asks for, and it's the part that holds. Each answer is at the end of the lesson, and the worked examples above cover identifying and evaluating, including the signed case.
Identify each as an expression or an equation:
Reveal answerHide to problem 1
expressionReveal answerHide to problem 2
equationReveal answerHide to problem 3
expressionReveal answerHide to problem 4
equationEvaluate each expression at the given value:
Reveal answerHide to problem 5
14Reveal answerHide to problem 6
7Reveal answerHide to problem 7
11Reveal answerHide to problem 8
-8Reveal answerHide to problem 9
-13Reveal answerHide to problem 10
4Reveal answerHide to problem 11
5Reveal answerHide to problem 12
12(Problems 11 and 12 are the ones to slow down on: 11 is a negative under an exponent, where (−2)² = 4, not −4, and 12 shows 2(x + 3) is not 2x + 3.)
Negatives meet order of operations: now that negatives from Lesson 1.5 are in hand, mix them back into the order of operations from Lesson 1.3. This is a common slip in early algebra, so it's worth a careful look. Watch the two cases side by side.
Without parentheses, −2² means −(2²): the exponent binds only to the 2, and the leading minus is applied after squaring, so −2² = −(4) = −4. With parentheses, the whole −2 is squared: (−2)² = (−2)(−2) = 4.
Say it as "no parentheses, square first, then take the opposite." Read aloud, the first is "the negative of two squared," the second is "the quantity negative two, squared."
Evaluate (mind the sign and the exponent's reach):
- −2²
- (−2)²
- −3²
- −2² + 1
- 10 − 2·(−3)
- 8 + (−4)/2
AnswersTry each one yourself first, then open to check.
- -4 14. 4 15. -9 16. -3 17. 16 18. 6
(In 13 vs 14 the only change is the parentheses, and it flips the sign of the answer; 17 and 18 drop a negative into the multiply/divide step. If you land on −2² = 4, you squared before applying the minus, so come back to "the exponent only touches the 2.")