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Solving via Substitution: The Spy Method

Replacing Variables to Crack the Code of Systems

Imagine you have a puzzle with two mystery numbers. You know that "Number A" is exactly 5 more than "Number B." You also know that their total is 15. How do you find them? You intuitively use substitution: you replace "Number A" with the idea of "Number B + 5."

In algebra, Solving via Substitution is one of the most powerful algebraic techniques for solving Systems of Equations. It allows you to take a complex system with two variables (x and y) and turn it into a simple equation with just one.

1. The Core Concept

The logic is simple: If y is equal to a group of apples, you can replace y with that group of apples anywhere you see it.

If we know that y = 3x, then "y" and "3x" are interchangeable. We can take "3x" and plug it into any other equation in place of "y."

2. The 3-Step Strategy

To solve a system using substitution, follow this roadmap:

  1. Isolate: Get one variable by itself in one equation (e.g., x = ... or y = ...).
  2. Substitute (The Swap): Plug that expression into the other equation. Now you have an equation with only one variable. Solve it.
  3. Plug Back In: Take the number you found and plug it back into the first equation to find the second variable.

3. Example 1: The Easy Case

Sometimes, the variable is already isolated for you.

System:
1) y = 2x + 1
2) 3x + y = 11

Step A: Substitute

Since y is "2x + 1", we replace y in equation #2.
3x + (2x + 1) = 11

Step B: Solve

Combine like terms:
5x + 1 = 11
5x = 10
x = 2

Step C: Find the other half

Plug x=2 back into equation #1.
y = 2(2) + 1
y = 4 + 1
y = 5

Solution: (2, 5)

4. Example 2: The Rearrangement Case

Sometimes you have to do a little work before you can substitute.

System:
1) x - 2y = 3
2) 4x + 3y = 23

Step A: Isolate

Look at equation #1. It is easy to get x alone by adding 2y to both sides.
x = 2y + 3

Step B: Substitute

Plug (2y + 3) into equation #2 where x used to be.
4(2y + 3) + 3y = 23

Step C: Solve

Distribute the 4 (Don't forget this step!):
8y + 12 + 3y = 23
11y + 12 = 23
11y = 11
y = 1

Step D: Finish It

Plug y=1 back into our isolated equation (x = 2y + 3).
x = 2(1) + 3
x = 5

Solution: (5, 1)

5. The Danger Zone: Parentheses

The most common mistake students make is forgetting parentheses during substitution. This leads to distribution errors.

Wrong: 4 * 2y + 3 + 3y = 23
Right: 4 * (2y + 3) + 3y = 23

Always wrap your substituted expression in parentheses to ensure you multiply everything correctly.

6. When to Use Substitution vs. Elimination

While both methods work, Substitution is usually faster when:

  • One variable has a coefficient of 1 or -1 (e.g., x + 3y = 10).
  • One equation is already solved for a variable (e.g., y = 5x).

If you see a mess of numbers like 3x + 4y = 10 and 5x - 2y = 7, the Elimination method might be easier.

7. Conclusion

Solving via substitution is like being a detective. You find a clue (what x equals), and you use that clue to crack the rest of the case. It transforms a two-variable problem into a one-variable problem, making the impossible solvable. Remember: Isolate, Substitute, Solve.