Questions: Variables and Expressions Review

Question 1

A student is given the expression 3x + 2y − x + 5 and told to 'solve for x.' What is the fundamental problem with this instruction?

• A. There is no problem — the student should isolate x on one side of the expression
• B. The problem has no solution because there are two variables, which is not allowed in Algebra 1
• C. 'Solving' does not apply to expressions — expressions are simplified, not solved; solving requires an equation with an equals sign
• D. The student must first apply the distributive property before any other steps

Answer: (C)

This is the core distinction: expressions are simplified (rewritten in equivalent form with fewer terms), while equations are solved (a value of the variable is found that makes both sides equal). An expression like 3x + 2y − x + 5 has no equals sign, so it cannot be 'solved.' It can be simplified to 2x + 2y + 5. The right action depends entirely on which type of mathematical object you're working with.

Question 2

A student simplifies 4x + 3y − x and writes 6xy as the answer. Which error did they make?

• A. They forgot to apply the distributive property before combining terms
• B. They treated 4x and 3y as like terms and combined them — but unlike terms cannot be combined
• C. They applied the wrong order of operations, adding before subtracting
• D. They dropped the negative sign when subtracting the final x term

Answer: (B)

Like terms must have identical variable parts — same variables, same exponents. The terms 4x and 3y have different variables (x vs. y), so they are unlike terms and cannot be combined. The correct simplification is 4x − x + 3y = 3x + 3y. Writing 6xy confuses addition (which preserves the variable parts) with multiplication (which would produce xy). This is one of the most persistent errors in early algebra.

Question 3

The expression 5a − 3b + 2a can be correctly simplified to 7a − 3b.

• T. True
• F. False

Answer: True

5a and 2a are like terms — they share the same variable (a) with the same exponent (1). Combining them: 5a + 2a = 7a. The term −3b has a different variable and cannot be combined with the a-terms. The simplified form is 7a − 3b. This is a straightforward application of combining like terms.

Question 4

Distributing the negative sign in −(2x − 4) gives −2x − 4.

• T. True
• F. False

Answer: False

The negative multiplies every term inside the parentheses. −(2x − 4) = −1 · 2x + (−1) · (−4) = −2x + 4. The second term's sign flips because negative times negative equals positive. Writing −2x − 4 is the most common error when distributing negatives. The correct result is −2x + 4.

Question 5

Why can't you 'solve' the expression 3x + 7? What operation applies to it instead, and what additional information would you need to find a specific numerical value for x?

Model answer: An expression has no equals sign, so there is no constraint pinning down the value of x. You can simplify an expression — rewrite it in a cleaner or shorter equivalent form — but 'solving' requires an equation. To find a specific value for x, you would need either an equation (e.g., 3x + 7 = 22, which you can solve to get x = 5) or a specific value to substitute for x (e.g., if x = 4, then 3(4) + 7 = 19).

The expression vs. equation distinction determines which algebraic operations are even meaningful. Confusing them leads to nonsense: 'solving' an expression, or trying to 'simplify' an equation without maintaining the equality. Every algebra problem starts with identifying what you have — expression or equation — because that determines what you do with it.