Proving the Elimination Method Works

Why Every Elimination Step Is Valid

In this lesson:

• State and prove the system-equivalence theorem
• Apply the theorem to justify elimination

What You Will Be Able to Do

By the end of this lesson, you should be able to:

1. State the system-equivalence theorem (REI.C.5)
2. Prove both directions of the theorem
3. Explain why elimination is justified by this theorem
4. Identify legal and illegal operations on systems
5. Connect proof technique to properties of equality

What the Elimination Step Looks Like

System: and

Add the equations (k = 1):

New system: and

Does the new system have the same solution?

One Example Is Not a Proof

We verified the theorem for one specific system — but the theorem claims to hold for:

• Any two-equation system (any coefficients, any constants)
• Any value of
• All solutions (a system may have infinitely many)

To prove the theorem, we must argue for the general case.

The Theorem We Are Proving

Theorem (REI.C.5): Given a system of two equations:

Replace with . New system:

has the same solution set as the original.

Proving Equivalence: Two Directions Required

To prove "same solution set," show both:

Every solution of original satisfies new system

Every solution of new system satisfies original

Direction 1: Original Solutions Satisfy the New System

Let be any solution of the original system. Then:

Does satisfy ?

Also satisfies (unchanged). So satisfies the new system.

"Any Solution" Makes the Proof General

Proving with a specific only covers one solution.

"Let be any solution" covers all solutions at once.

• holds for any — so it holds for every solution

This is how mathematics proves universal statements.

Are We Done After Direction 1?

Direction 1 proves: every original solution satisfies the new system.

Does this mean the systems are equivalent?

Not yet. The new system might have solutions the original doesn't.

We need Direction 2: every new solution must also satisfy the original system.

Direction 2: New Solutions Satisfy the Original System

Let satisfy the new system. Then:

Does satisfy ?

Key move:

The Key Algebraic Move in Direction 2

This is just algebra — adding and subtracting .

But why does this help?

• (given — new equation is satisfied)
• (given — second equation is unchanged)
• So both terms on the right are zero

Both Directions Complete the Proof

Both directions established:

: Every original solution satisfies the new system

: Every new solution satisfies the original system

Conclusion: The two systems have exactly the same solution set.

Each elimination step you have ever performed was justified by this proof.

Cite the Theorem at Each Elimination Step

Solve: (E₁) and (E₂)

Step 1 (k = −3): replace E₁ with E₁ + (−3)·E₂

 (by REI.C.5)

Step 2: Substitute into E₂ and solve for .

Legal and Illegal Operations on Systems

Illegal operations may change the solution set — no theorem covers them.

Illegal Operation: Find the Counterexample

Modify to by adding 3 to the right only.

: does it satisfy ?

. ✗ The solution set changed.

This is not covered by REI.C.5.

Complete the Direction 2 Proof

Fill in each blank and give the reason:

Let satisfy the new system. Then both equations evaluate to [blank].

Using : $E_1(x_0, y_0) = $ [blank] [blank] $= $ [blank].

Therefore satisfies the original system.

Classify These Operations: Legal or Illegal?

For each, write "Legal (REI.C.5, k = ___)" or "Illegal (counterexample: ___)"

1. Replace with
2. Replace with
3. Replace with
4. Add 5 to both sides of only
5. Swap and

Solve a System and Cite the Theorem

Solve by elimination and cite the theorem at each step:

For each replacement step, write: "By REI.C.5 with k = ___, replacing with produces an equivalent system."

What You Can Do Now

✓ State REI.C.5: gives an equivalent system

✓ Prove both directions with an arbitrary

✓ Cite the theorem to justify each elimination step

Both directions required — Direction 1 alone is insufficient

Only theorem-authorized operations are legal

Coming Up: Using Justified Elimination

Next: HSA.REI.C.6 — solve systems of two linear equations in two variables.

You now know why elimination works.

REI.C.6 shifts focus to how to apply it efficiently — including when to use substitution instead.