Substitution and Elimination Methods

In this lesson:

• Solve systems exactly using substitution
• Solve systems exactly using elimination
• Choose the most efficient method for any given system

Learning Objectives for This Lesson

By the end of this lesson, you will:

1. Solve systems using substitution, verified in both equations
2. Solve systems using elimination, scaling when needed
3. Recognize the algebraic signals for special cases
4. Choose the most efficient method for any system

Moving from Geometry to Algebraic Methods

In Lesson 1: graphing finds the intersection visually — approximate for non-integer answers.

Today: algebraic methods find the intersection exactly — regardless of the coordinate values.

Substitution: reduce a 2-variable problem to a 1-variable equation

Elimination: add equations to make one variable disappear

Substitution Method: The Five Steps

1. Isolate one variable (prefer coefficient 1 or )
2. Substitute into the other equation
3. Solve the one-variable equation
4. Back-substitute to find the second variable
5. Verify in both originals

Back-substitute into an original — not a simplified version.

Substitution Example 1: Integer Solution

System: and

1. Isolate:
2. Substitute:
3. Solve: , so
4. Back-sub:
5. Verify in both originals ✓

Substitution Example 2: Clean Integer Case

System: and

1. is already isolated
2. Substitute:
3. Solve: , so
4. Back-sub: . Verify ✓

Solution:

Substitution Applied to Special Case Systems

False statement → no solution

, : — no solution

True identity → infinite solutions

, : — infinite solutions

Guided Practice: Applying the Substitution Method

Solve: and

Which variable is easiest to isolate? In which equation?

Work through all 5 steps, then advance

Check-In: Recognizing When to Use Substitution

Which system is best solved by substitution?

• A: and
• B: and
• C: and

Elimination Method: The Six Steps

1. Align in standard form ()
2. Choose a variable to eliminate
3. Scale so chosen coefficients are opposites
4. Add — the chosen variable cancels
5. Solve the resulting equation
6. Back-substitute and verify in both originals

Visual Intuition Behind Elimination Method

When the coefficients are and , adding cancels completely.

The goal: engineer opposite coefficients before adding.

Elimination Example 1: Direct Cancel

System: and

coefficients and cancel directly:

Add:

Back-sub:

Verify: ✓; ✓. Solution:

Elimination Example 2: Scaling Required

System: and

Scale to cancel : multiply by 2 and 3 respectively.

Add: , so . Back-sub: .

Method Selection: Which to Choose?

Quick rule: substitution when one variable has coefficient 1; elimination when coefficients share a pattern.

Comparing Both Methods with Examples

Practice: Choose a Method and Solve

Solve each using the most efficient method:

1. and
2. and
3. and

Answers to Method Selection Practice

1. Elimination: add → , , . Solution:

2. Elimination (scale by 2 and 5): , , . Solution:

3. Rewrite: — same line. Infinite solutions.

Key Takeaways from This Lesson

• Substitution: isolate → substitute → solve → back-sub → verify
• Elimination: scale for opposites → add → solve → back-sub → verify
• False statement → no solution; true identity → infinite solutions
• Coefficient 1 → substitution; opposites → elimination

What's Next: Linear and Quadratic Systems

Next lesson: HSA.REI.C.7 — Solve systems involving one linear and one quadratic equation

Substitution will be your primary tool — exactly what you practiced today.