Simultaneous Equations: Equal Values and Substitution Method

The Substitution Method is a technique for solving systems of equations by expressing one variable in terms of the other and substituting it into the second equation.

• Step 1: Express one equation in terms of y
  Solve for $y$ in one equation:

$$
\begin{cases}
2x + y = 8 \\
x – y = 1 &\Rightarrow y = x – 1
\end{cases}
$$

• Step 2: Substitute $y = x – 1$ into the other equation

$$2x + (x – 1) = 8 \Rightarrow x = 3$$

• Step 3: Find $y$ by substituting $x$
  Substitute$x = 3$ back in $y = x – 1$ gives:

$$y = 3 – 1 = 2$$

• Step 4: Verify the solution
  The final solution is:

$$ \{ x = 3, \quad y = 2 \} $$

Both methods always yield the same solution! However, one method may be simpler than the other, depending on the given equations.

Equal Values Method
This method is useful when isolating the same variable in both equations is easy.

🔹 Example:

$$
\begin{cases}
2x = 4y – 2 &\Rightarrow x = 2y – 1 \\
x – y = 2 &\Rightarrow x = y + 2
\end{cases}
$$

Since both are solved for \(x\), you can directly set them equal:

$$ 2y – 1 = y + 2 $$

This simplifies quickly to \( y = 3 \), making it an efficient method.

Substitution Method
This method is better when one equation is already solved for a variable or can be easily rearranged.

🔹Example:

$$
\begin{cases}
2x + 3y = 3 \\
3y = 2x – 1
\end{cases}
$$

Since the second equation is already solved for \(3y\), substituting it into the first equation:

$$ 2x + (2x – 1) = 3 $$

This quickly leads to \( x = 1 \).