Simultaneous Equations: Equal Values and Substitution Method

The Equal Values Method is a technique for solving systems of equations by isolating the same variable in both equations and setting them equal to each other.

• Step 1: Express both equations in terms of y
  Rearrange each equation to solve for y:

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

• Step 2: Set the two new expressions equal
  Since both are equal to y, we set them equal to each other:

$$ 8\, -\, 2x = x\, -\, 1 $$ $$ x = 3 $$

• Step 3: Find y by substituting x
  Substituting $ x = 3 $ into any of the equations, preferably a simpler one, gives:

$$ y = 3\, -\, 1 = 2 $$

• Step 4: Verify the solution
  The final solution to the system of equations is:

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

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.

🍀 When is the Equal Values Method Useful?

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

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

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

$$ 2y\, -\, 1 = y + 2 $$

• This simplifies quickly to \( y = 3 \).

🍀 When is the Substitution Method Useful?

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

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

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

$$ 2x + (2x\, -\, 1) = 3 $$

• This quickly leads to \( x = 1 \).