Lesson 7.4

Substitution (Level 2)

Sometimes the equations aren't ready for us. We have to do a little prep work (isolating a variable) before we can substitute.

Introduction

In Level 1, we saw . But what if we see ? We can't substitute yet. We need to isolate one variable first to create our "blob."

Past Knowledge

Lesson 2.15 (Literal Equations). You need to be comfortable moving terms to get a letter by itself.

Today's Goal

Identify the easiest variable to isolate, rewrite the equation, and then solve.

Future Success

Choosing the "easy" path is a math skill. We want to avoid fractions if possible.

Key Concepts

The "Lonely Variable" Strategy

When choosing which variable to isolate, look for the one with a coefficient of 1 or -1.

Good Choice

The has no number in front. Just add and you are done.

Bad Choice

If you solve for , you'll have to divide by 5, creating messy fractions like .

Worked Examples

Example 1: Isolating X

Basic

Solve the system.

Step 1: Isolate

The first is "lonely". Move .

Step 2: Substitute

Plug into the other equation.

Step 3: Solve

Step 4: Boomerang

Use your rewritten equation:

Solution:

Example 2: Isolating Y

Intermediate

Solve the system.

Step 1: Isolate

Evaluate the first equation. Move .

(Divide by -1)

Step 2: Substitute & Solve

Step 3: Boomerang

Solution:

Example 3: Dealing with Fractions

Advanced

Solve the system.

Step 1: Isolate (Harder)

Nothing is "lonely". But in Eq 1, everything divides by 2.

(Divide by 2)

Step 2: Substitute & Solve

Conclusion

.

Solution:

Common Pitfalls

Plugging into the Wrong Spot

If you isolate , make sure you plug that blob into the spot in the OTHER equation. Don't plug it into !

Sign Errors when Isolating

When moving to the other side, it becomes . Many students forget to flip the sign.

Real-Life Applications

Resource Allocation:

  • A factory has limited "Machine Time" and limited "Man Hours".
  • You usually know relationships like "Man Hours = 2 times Machine Time".
  • This "substitution" logic allows managers to calculate exact production schedules.

Practice Quiz

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