Problem 38 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the equation (if possible). $$3=2+\frac{2}{z+2}$$

Step-by-Step Solution

Answer

The solution for the equation \(3=2+\frac{2}{z+2}\) is \(z = 0\).

Step 1: Identify the Original Equation

The given equation is \(3=2+\frac{2}{z+2}\). The objective is to solve for \(z\).

Step 2: Subtract 2 from Both Sides

To remove the constant on the right hand side, subtract 2 from both sides to make the equation \(1=\frac{2}{z+2}\)

Step 3: Inverse Multiplication

The right hand side is in fraction form. To eliminate the fraction, multiply both sides by \(z+2\). This leads to \(z+2=2\).

Step 4: Isolate the Variable

The variable \(z\) is not isolated. Subtract 2 from both sides to make \(z=2-2\).

Step 5: Calculate the Value of Variable

By performing the subtraction operation, it is found that \(z=0\).

Key Concepts

Fraction Elimination

When solving algebraic equations, especially those containing fractions, the first step is often to eliminate the fraction to simplify the equation. Fractions can sometimes make an algebraic equation seem more complicated than it actually is. In the equation \(3=2+\frac{2}{z+2}\), the fraction \(\frac{2}{z+2}\) can be intimidating.
To eliminate this fraction, you'll want to perform an operation that will transform it into a simpler form. In this case, multiplying both sides of the equation by \(z+2\) does the trick. Here's why:

• Multiplying by \(z+2\) cancels out the \(z+2\) in the denominator on the right-hand side.
• This operation turns the equation into \(z+2 = 2\), which is much easier to solve.

By removing the fraction, you've made the equation more manageable and ready for simple operations to isolate the variable.

Variable Isolation

Once the equation \(z+2=2\) is obtained, the next step involves isolating the variable you're solving for, which is \(z\) in this case. The concept of variable isolation means rearranging the equation such that the variable is alone on one side of the equation. This process makes it possible to determine the variable's value. Here’s how it works here:
To isolate \(z\), subtract 2 from both sides of \(z+2=2\) to undo the addition of 2. This yields the equation \(z = 2 - 2\), simplifying to \(z=0\).

• Performing the same operation on both sides of the equation maintains its balance.
• Isolating \(z\) makes it easy to see that \(z\) equals 0.

Variable isolation is a crucial step in algebra. It allows you to focus directly on finding the value you need, one operation at a time.

Inverse Operations

Inverse operations are the mathematical processes that 'undo' each other; they are key to solving equations. Common pairs of inverse operations include addition and subtraction, as well as multiplication and division. By using inverse operations, you can effectively work toward isolating variables or simplifying expressions.
In the equation \(1 = \frac{2}{z+2}\), by applying the inverse operation of division (which is multiplication), you can eliminate the fraction. Multiplying both sides by \(z+2\) is an example of using inverse operations to remove the fraction from the equation. Later, subtracting 2 from both sides of the equation \(z+2=2\) is another application of inverse operations to isolate \(z\).

• Multiplication undoes division by removing the denominator.
• Subtraction undoes addition, helping isolate the variable.

Utilizing inverse operations is like peeling away layers of an equation to reveal the solution hidden beneath. They're essential tools for simplifying and solving equations efficiently.