Problem 3 - Algebra A Combined Function

Question

Solve each equation. Check each solution. See Examples 1 through \(6 .\) \(3 x=0\)

Step-by-Step Solution

Answer

The solution is \(x = 0\).

Step 1: Identify the Equation

The given equation is a simple linear equation: \(3x = 0\). Our task is to find the value of \(x\) that satisfies this equation.

Step 2: Isolate the Variable

In order to isolate the variable \(x\), we need to divide both sides of the equation by the coefficient of \(x\), which is 3. This gives us \(x = \frac{0}{3}\).

Step 3: Simplify the Equation

Simplifying \(x = \frac{0}{3}\), we find \(x = 0\) since any number divided by a non-zero number is zero.

Step 4: Check the Solution

To verify our solution, substitute \(x = 0\) back into the original equation: \(3 \times 0 = 0\), which confirms that the left side equals the right side.

Key Concepts

Isolating the Variable

When solving linear equations, one of the first steps is to isolate the variable you want to solve for. This means getting the variable by itself on one side of the equation. In our example, the equation is \(3x = 0\).
To isolate \(x\), we must eliminate the coefficient that is accompanying it. Since the coefficient here is 3, we can "undo" this multiplication by performing the inverse operation: division. Therefore, we divide both sides of the equation by 3.
This results in \(x = \frac{0}{3}\), which simplifies to \(x = 0\).
When performing operations to isolate the variable, remember:

• Choose operations that cancel out the constants or coefficients on the variable's side.
• Always perform the same operation on both sides of the equation to maintain balance.
• Think of the equation like a balance scale; whatever you do to one side must be done to the other.

Checking Solutions

After finding a value for the variable, it's vital to check that this solution is correct. You do this by substituting your solution back into the original equation.
For our equation \(3x = 0\), we found \(x = 0\). Let's verify by substituting:\
Substitute \(x = 0\) into the equation:

• Left side of the equation: \(3 \times 0 = 0\)
• Right side of the equation: 0
• Both sides are equal, confirming \(x = 0\) is indeed the correct solution.

Checking solutions ensures that there were no mistakes during calculations, especially in more complex equations. It's a reliable way to guarantee the accuracy of your solution.

Simplifying Equations

Simplifying is an important part of solving equations. It involves reducing the equation to its simplest form to make it easier to analyze and solve.
In our example, \(x = \frac{0}{3}\) simplifies directly to \(x = 0\).
Here are some tips for simplifying equations:

• Always perform operations in a way that reduces unnecessary complexity.
• Combine like terms where possible. For instance, if the equation had been \(3x + 0 = 0\), we might first combine like terms to have \(3x = 0\).
• Look for common factors that can be divided out.
• Sometimes simplification involves eliminating denominators by multiplying both sides of the equation, but always check that the denominator isn't zero.

Simplifying equations is a key step in ensuring the solutions we derive are both correct and presented in the most interpretable form.