Problem 4 - Algebra A Combined Function

Question

Solve each equation. Check each solution. See Examples 1 through \(6 .\) \(2 x=0\)

Step-by-Step Solution

Answer

The solution to the equation is \(x = 0\).

Step 1: Isolate the Variable

To solve the equation \(2x = 0\), we need to isolate \(x\). We'll do this by dividing both sides of the equation by 2. This gives us: \(x = \frac{0}{2}\).

Step 2: Simplify the Equation

Next, simplify the right side of the equation. Since \(0\) divided by any non-zero number is \(0\), we have \(x = 0\).

Step 3: Check the Solution

To verify our solution, substitute \(x = 0\) back into the original equation: \(2 \times 0 = 0\). Since both sides of the equation are equal \(0 = 0\), the solution is correct.

Key Concepts

Isolating the Variable

In solving linear equations, isolating the variable is a crucial first step. It essentially means getting the variable you are solving for on one side of the equation by itself. This helps us find the value of that variable easily.
To isolate a variable, you often need to perform operations that involve addition, subtraction, multiplication, or division. The goal is to undo operations around the variable.
For example, consider the equation \(2x = 0\). Here, the variable \(x\) is attached to the number 2 by multiplication. To isolate \(x\), we need to do the opposite of multiplication, which is division.

• Divide both sides of the equation by 2: \(\frac{2x}{2} = \frac{0}{2}\)
• This simplifies to \(x = 0\)

Now, \(x\) is by itself, which is the isolated variable on the left side of the equation.

Verifying Solutions

After solving an equation, it's always a good practice to verify your solution. This step ensures that the value obtained for the variable satisfies the original equation.
Verification involves substituting the found value back into the original equation and checking if the left and right sides are equal.
For the equation \(2x = 0\), we found that \(x = 0\). To verify:

• Substitute \(x = 0\) back into the original equation: \(2 \times 0 = 0\)
• Check if the computations on both sides of the equation result in the same value.

In this case, both sides equal 0, confirming that \(x = 0\) is indeed the correct solution. Verification helps to cement your understanding and ensures that no mistakes were made in the calculation process.

Simplifying Equations

Simplifying equations is an important part of solving linear problems. Simplification involves performing arithmetic operations so that the equation becomes as simple as possible. This typically means reducing fractions, combining like terms, or eliminating coefficients.
In our example, after isolating the variable, we have \(x = \frac{0}{2}\). Simplifying this gives us \(x = 0\) because any number divided by a non-zero number remains zero.
Taking time to simplify equations ensures accuracy in mathematical operations and provides an easily understandable final result.

• Simplification reduces complexity, making the solution clearer.
• It helps in checking the correctness of the solution later during verification.

Always remember, simplified results contribute significantly to the clarity when presenting or using these results in further calculations.