North CarolinaMathsSyllabus dot point

Why does replacing one equation with a combination of the two keep the same solution?

Prove that replacing one equation in a system with the sum of it and a multiple of the other produces an equivalent system (NC.M1.A-REI.5).

An NC Math 1 EOC answer on equivalent systems (NC.M1.A-REI.5): why adding a multiple of one equation to another preserves the solution set, which is the justification behind the elimination method.

Generated by Claude Opus 4.89 min answerUpdated 2026-06-13

Reviewed by: AI editorial process; not yet individually human-reviewed

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What this topic is asking
What "equivalent" means
Why a combination keeps the solutions
Connecting to elimination
How the NC Math 1 EOC examines this topic
Why reversibility is the safeguard
Try this

What this topic is asking

NC.M1.A-REI.5 asks you to prove that, given a system of two linear equations, replacing one equation by the sum of that equation and a multiple of the other produces an equivalent system, one with the same solution set. This is the reasoning that makes the elimination method valid, so the standard is about understanding why a procedure works, not just doing it.

What "equivalent" means

The key word is solution set.

Why a combination keeps the solutions

The proof is short and worth knowing.

Why elimination is valid

Show that replacing equation 2 with (equation 1 added to equation 2) keeps the same solution.

step 1 Start with a common solution

Suppose $(x, y)$ satisfies both $E_1$ and $E_2$ , so both are true statements at that point.

step 2 Combine true statements

Adding two true equations gives a true equation: $E_1 + E_2$ holds at $(x, y)$ . So $(x, y)$ also satisfies the new pair $\{E_1,\ E_1 + E_2\}$ .

step 3 Show no solutions are gained

The step is reversible: from $E_1$ and $E_1 + E_2$ you recover $E_2$ by subtracting $E_1$ . So any solution of the new system also solves the original.

step 4 Conclude

The two systems have identical solution sets, so they are equivalent.

Connecting to elimination

Elimination uses exactly this principle. When you add equations (after scaling one so a variable cancels), you are replacing an equation with a combination of the two. Because the combination keeps the solution set, the value you find is genuinely the system's solution. The standard explains why the trick of "make the $y$ terms opposite, then add" does not distort the answer.

How the NC Math 1 EOC examines this topic

Multiple choice. Choose the statement that correctly explains why a step keeps the solution.
Technology-enhanced. Select all equations equivalent to a given system, or order steps of a valid elimination.
Short reasoning. Identify why scaling and adding is allowed.

This standard is the logical backbone of solving systems algebraically, and it parallels the reversibility argument behind solving linear equations, where each step preserves the solution.

Why reversibility is the safeguard

The reason equivalent-system moves are trustworthy is that they can be undone. Multiplying an equation by a nonzero constant can be reversed by dividing; adding a multiple of one equation to another can be reversed by subtracting it back. Reversibility guarantees two things at once: no solution is lost (each original solution survives the move) and no solution is invented (each new solution traces back to the originals). That two-way street is precisely what "same solution set" requires. The same logic explains why you must use a nonzero multiplier, multiplying by zero is not reversible and would erase information. Understanding reversibility turns elimination from a memorized trick into a justified method.

Try this

Q1. The solution of a system is $(2, 5)$ . Does it satisfy the sum of the two equations? [1 point]

Cue. Yes. A point that satisfies both equations satisfies any sum or multiple of them.

Q2. Why must the multiplier in "multiply then add" be nonzero? [1 point]

Cue. Multiplying by zero is not reversible and erases the equation, so it would not give an equivalent system.

Exam-style practice questions

Practice questions written in the style of NCDPI exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

NC Math 1 EOC (style)2 marksGiven

x + y = 6

and

2x - y = 3

, show that adding the two equations gives an equation that the solution still satisfies.

Show worked answer →

Adding gives $3x = 9$ , so $x = 3$ , and the solution $(3, 3)$ satisfies it.

The solution of the system is $(3, 3)$ (since $x + y = 6$ and $2x - y = 3$ both hold). Adding the equations: $(x + y) + (2x - y) = 6 + 3$ , giving $3x = 9$ , so $x = 3$ . Because $(3, 3)$ makes both original equations true, it also makes their sum true. This is the A-REI.5 idea: a combination of the equations keeps the same solution.

NC Math 1 EOC (style)1 marksWhy can you multiply one equation of a system by a constant before adding it to the other? (A) it changes the solution (B) it keeps the same solution set (C) it removes a variable forever (D) it is not allowed

Show worked answer →

The correct answer is (B), it keeps the same solution set.

Multiplying an equation by a nonzero constant produces an equivalent equation with the same solutions, and adding a multiple of one equation to another produces an equivalent system. The solution that satisfied both originals still satisfies the new pair. This is the justification for scaling in the elimination method.

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Sources & how we know this

North Carolina Standard Course of Study for Mathematics — NC Department of Public Instruction (2024)
EOC NC Math 1 and NC Math 3 Test Specifications — NC Department of Public Instruction (2024)