91Ó°ÊÓ

In solving algebraic equations, the substitution method is a crucial technique. This method involves replacing the variable in the equation with a specific value to check if it makes the equation true. Let's talk about this in the context of our exercise.

Consider the equation given: \[x^2 - 5x = 15 - 3x\]

We need to determine if the values \(x = 0\) and \(x = -3\) satisfy this equation. To do this, we perform substitution:

* **For \(x = 0\):**
Substitute 0 into the equation: \[0^2 - 5(0) = 15 - 3(0)\]
Simplify both sides:
\[0 = 15\]
Since 0 does not equal 15, \(x = 0\) does not satisfy the equation.

* **For \(x = -3\):**
Substitute -3 into the equation: \[(-3)^2 - 5(-3) = 15 - 3(-3)\]
Simplify both sides:
\[9 + 15 = 15 + 9\]
\[24 = 24\]
Since both sides are equal, \(x = -3\) does satisfy the equation. This method of substitution is particularly useful in verifying potential solutions for algebraic equations.

Simplifying equations is a fundamental step in solving them. It makes the equations easier to work with and understand.

Let's revisit our exercise to see how this is done:

• Start with the given equation: \[x^2 - 5x = 15 - 3x\]
  
• When substituting a value for \(x\), simplify both sides of the equation:
  
  * **For \(x = 0\):**
  Substitute and simplify:
  \[0^2 - 5(0) = 15 - 3(0)\]
  \[0 = 15\]
  As observed, the left side simplifies to 0, while the right simplifies to 15.
  
  * **For \(x = -3\):**
  Substitute and simplify:
  \[(-3)^2 - 5(-3) = 15 - 3(-3)\]
  \[9 + 15 = 15 + 9\]
  \[24 = 24\]
  The left side simplifies to 24, and the right side to 24.
  
  By simplifying each side of the equation, we can easily check if a given value of \(x\) maintains the equality.

Once you have simplified the equation using substitution, it is essential to verify if the obtained solutions are correct. Verification eliminates any doubts about the validity of the solutions found.

In our exercise:
* We substituted \(x = 0\) and found that it did not satisfy the equation:
\[0^2 - 5(0) eq 15 - 3(0)\]
\[0 eq 15\] This clearly shows \(x = 0\) is incorrect.

* We substituted \(x = -3\) and both sides simplified correctly:
\[(-3)^2 - 5(-3) = 15 - 3(-3)\]
\[24 = 24\] This indicates \(x = -3\) is indeed a solution.

Verification is a final check to ensure the solution is right. It helps build confidence in your work and confirms that the process followed was accurate.