Questions: Systems of Equations — Substitution Method

Question 1

Given the system y = 3x − 2 and 2x + y = 13, which step correctly begins the substitution process?

• A. Add the two equations together to eliminate y
• B. Substitute 3x − 2 in place of y in the second equation: 2x + (3x − 2) = 13
• C. Substitute 3x − 2 in place of y in the first equation: (3x − 2) = 3x − 2
• D. Solve the second equation for x first before doing anything else

Answer: (B)

Since the first equation already isolates y, we use it to replace y in the OTHER equation. Substituting into the same equation (Option C) just gives a tautology and makes no progress. Option A describes elimination, a different method. The whole point of substitution is to reduce the two-equation system to one equation in one unknown: 2x + (3x − 2) = 13 can be solved directly.

Question 2

A student isolates x from x − y = 3 to get x = y + 3, then substitutes into 2x + y = 9. They write: 2y + 3 + y = 9. What error did they make?

• A. They should have substituted into the first equation, not the second
• B. They forgot to multiply y + 3 by 2 — it should be 2(y + 3) + y = 9
• C. They should have isolated y instead of x
• D. They made no error — 2y + 3 + y = 9 is correct

Answer: (B)

When substituting x = y + 3 into 2x + y = 9, the entire expression (y + 3) replaces x, but the coefficient 2 applies to the whole expression: 2(y + 3) + y = 9, giving 2y + 6 + y = 9. Writing 2y + 3 + y treats the 2 as applying only to y but not the constant +3. This missing-parentheses error is the most common mistake in substitution and produces an incorrect value for y that will fail the check step.

Question 3

After substituting and simplifying, a student gets the equation 0 = 5. This means the system has infinitely many solutions.

• T. True
• F. False

Answer: False

A false statement like 0 = 5 means the system has NO solution — the two lines are parallel and never intersect. Infinitely many solutions occur when substitution produces a true identity like 0 = 0, meaning the two equations describe the same line. Both cases arise when all variables cancel, but the truth value of the resulting constant equation distinguishes them: false constant = no solution, true constant = infinitely many solutions.

Question 4

The substitution method always produces an exact answer, unlike the graphing method.

• T. True
• F. False

Answer: True

Graphing finds the intersection visually, requiring you to read coordinates off a drawn grid — a process prone to estimation error, especially for non-integer solutions. Substitution is an algebraic process that produces exact rational (or irrational) values without any visual approximation. This is one of the main advantages of substitution over graphing, and why graphing is best used to understand the concept and check reasonableness rather than as a primary solving method.

Question 5

Explain why substituting an expression for a variable requires parentheses, using a specific example to show what goes wrong without them.

Model answer: When you substitute an expression like (x + 1) for y in an equation like 3 − y, the expression takes the place of the variable and the operation applies to the whole thing. With parentheses: 3 − (x + 1) = 3 − x − 1 = 2 − x. Without parentheses: 3 − x + 1 = 4 − x — a different expression. The subtraction distributes across all terms inside the parentheses, but without parentheses, only the first term gets the sign, silently dropping the sign change on all remaining terms.

This is the most mechanical but consequential step in substitution. The variable y has operations applied to it as a single unit; when you replace y with a multi-term expression, those operations apply to the whole expression. Parentheses are the notation that enforces this. Without them, only the leading term gets the operation, producing answers that cannot be verified when checked in both original equations — which is why the check step catches this error.