Problem 69 - Algebra 2

Question

Solve each equation. Check your solution. $$ 0=\frac{1}{2} x-3 $$

Step-by-Step Solution

Answer

The solution is \(x = 6\).

Step 1: Isolate the Variable Term

The equation given is \(0 = \frac{1}{2}x - 3\). To solve for \(x\), first add 3 to both sides to begin isolating the \(x\) term. This gives: \(3 = \frac{1}{2}x\).

Step 2: Clear the Fraction

Now, we want to eliminate the fraction in \(\frac{1}{2}x\). Multiply both sides of the equation by 2: \(2 \times 3 = 2 \times \frac{1}{2}x\), simplifying to \(6 = x\).

Step 3: Verifying the Solution

Substitute \(x = 6\) back into the original equation to ensure it satisfies the equation: \(0 = \frac{1}{2} \times 6 - 3\). Calculate the right side: \(\frac{1}{2} \times 6 = 3\), and \(3 - 3 = 0\). The left side equals the right side, confirming that the solution is correct.

Key Concepts

Isolating Variables

When solving linear equations, one essential step is getting the variable alone on one side of the equation. This is what mathematicians refer to as isolating the variable.
To start, you should look for the term containing the variable and perform inverse operations to move other numbers to the opposite side. In the exercise, we have the equation \(0 = \frac{1}{2}x - 3\).
Here, the term \(\frac{1}{2}x\) includes the variable \(x\). We can isolate this term by adding 3 to both sides of the equation, giving us \(3 = \frac{1}{2}x\).
This step is crucial in maintaining the equality of the equation while moving unwanted terms away from the variable term.
Remember, whatever operation you do to one side of the equation, you must do to the other side to keep the equation balanced. This process prepares the equation for the next steps: handling any coefficients or fractions attached to the variable.

Fractions in Equations

Fractions often make equations look complicated, but they can be managed systematically. In the context of an equation with fractions, like in our example, the goal is to "clear" the fraction.
The equation at this stage is \(3 = \frac{1}{2}x\). To eliminate the fraction, multiply both sides by the denominator of the fraction. Here, the denominator is 2.
When you multiply:

• Left side: \(2 \times 3 = 6\)
• Right side: \(2 \times \frac{1}{2}x = x\)

After simplifying, we have \(6 = x\). By multiplying both sides, the fraction is cleared, and the equation becomes a simple equality.
Clearing fractions early in the equation simplifies the process and reduces potential errors later on.
If there were multiple fractions, finding the least common multiple (LCM) can help to clear them out efficiently.

Verification of Solutions

After solving an equation, checking the solution is a vital step to avoid errors and confirm correctness. Verification involves substituting the solution back into the original equation and ensuring both sides of the equation are equal.
In our exercise, we found \(x = 6\). We substitute it back into the original equation:
\(0 = \frac{1}{2} \times 6 - 3\). Calculate step by step:

• \(\frac{1}{2} \times 6\) results in 3.
• 3 minus 3 equals 0.

The left side \(0\) matches the right side \(0\), meaning our solution is verified.
This step not only solidifies your understanding but also helps catch any mistakes made during calculation. Verifying solutions ensures accuracy and builds confidence in solving equations.