What Does 'Variable Already Isolated' Mean in Substitution Method?

📚 Understanding "Variable Already Isolated"

In the context of solving systems of equations using the substitution method, the phrase "variable already isolated" refers to a situation where one of the equations is written in the form $y = ...$ or $x = ...$. This means the variable (either $x$ or $y$) is by itself on one side of the equation, and the other side consists of an expression involving the other variable (and possibly constants).

📜 History and Background

The substitution method has been used for centuries as a way to simplify and solve systems of equations. The idea is to express one variable in terms of the other, allowing you to reduce the system to a single equation with a single variable. This equation can then be solved directly.

🔑 Key Principles

• 🔍 Definition: A variable is isolated when it stands alone on one side of the equation, with a coefficient of 1. For example, in the equation $y = 3x + 2$, $y$ is isolated.
• ➕ Advantage: When a variable is already isolated, you can directly substitute the expression it equals into the other equation. This simplifies the process and saves a step.
• ➖ Disadvantage (If not isolated): If neither variable is isolated, you must first manipulate one of the equations to isolate a variable. This involves algebraic operations like addition, subtraction, multiplication, or division.
• 🧮 Substitution: Once you've identified the isolated variable (or isolated one yourself), substitute the expression on the other side of the equals sign into the *other* equation wherever you see that variable.
• 🧩 Solving: After the substitution, you'll have an equation with only one variable. Solve for that variable.
• 🔄 Back-Substitution: Substitute the value you found back into the isolated variable equation to find the value of the other variable.
• ✅ Verification: Plug both values into the original equations to verify your solution.

💡 Real-world Examples

Example 1: Variable Already Isolated

Consider the system of equations:

$y = 2x + 1$

$3x + y = 10$

Here, $y$ is already isolated in the first equation. We can substitute $2x + 1$ for $y$ in the second equation:

$3x + (2x + 1) = 10$

Simplifying, we get:

$5x + 1 = 10$

$5x = 9$

$x = \frac{9}{5}$

Now, substitute $x = \frac{9}{5}$ back into the first equation:

$y = 2(\frac{9}{5}) + 1$

$y = \frac{18}{5} + 1 = \frac{23}{5}$

So, the solution is $x = \frac{9}{5}$ and $y = \frac{23}{5}$.

Example 2: Variable Not Isolated

Consider the system of equations:

$x + y = 5$

$2x - y = 1$

In this case, neither variable is isolated. We can isolate $y$ in the first equation:

$y = 5 - x$

Now, substitute $5 - x$ for $y$ in the second equation:

$2x - (5 - x) = 1$

Simplifying, we get:

$2x - 5 + x = 1$

$3x = 6$

$x = 2$

Substitute $x = 2$ back into the equation $y = 5 - x$:

$y = 5 - 2 = 3$

So, the solution is $x = 2$ and $y = 3$.

📝 Conclusion

Recognizing when a variable is already isolated simplifies the substitution method. It allows you to jump directly into substituting and solving, making the process more efficient and less prone to errors. If no variable is isolated, remember to make that your first step!