Problem 50 - Elementary Algebra Within Reach

Question

In Exercises 49 and 50 , justify each step of the equation. Then identify any properties of equality used to solve the equation. $$ \begin{aligned} \frac{4}{5} x &=4 x-16 \\ \frac{5}{4}\left(\frac{4}{5} x\right) &=\frac{5}{4}(4 x-16) \\ x &=5 x-20 \\ x-5 x &=5 x-20-5 x \\ x-5 x &=5 x-5 x-20 \\ -4 x &=-20 \\ \frac{-4 x}{-4} &=\frac{-20}{-4} \\ x &=5 \end{aligned} $$

Step-by-Step Solution

Answer

The step-by-step solution to the equation \(\frac{4}{5}x = 4x - 16\) involves utilizing primer properties of equality namely multiplication, subtraction, and division. The solution to the equation is \(x = 5\).

Step 1: Identifying the equality

Given the equation \(\frac{4}{5}x = 4x - 16\), it can be observed that there is an equality.

Step 2: Multiplying both sides by \(\frac{5}{4}\)

We multiply both sides by \(\frac{5}{4}\) to get rid of the fraction on the left side. The result is: \(x = 5x - 20\). The operation illustrates the multiplication property of equality.

Step 3: Rewriting equation

The equation is written as \(x - 5x = 5x - 20 - 5x\), then simplified as \(-4x = -20\). This demonstrates the subtraction property of equality.

Step 4: Dividing by coefficient

Both sides are then divided by -4 to isolate x. The equation is \(\frac{-4x}{-4} = \frac{-20}{-4}\), resulting in \(x = 5\). This shows the division property of equality.

Key Concepts

Properties of Equality

The properties of equality are crucial in solving equations because they allow us to perform operations on equations while maintaining their balance. Let's break down the main ones used in our exercise:

• Addition Property of Equality: This property allows us to add the same quantity to both sides of an equation without changing its equality.
• Subtraction Property of Equality: Similar to addition, we can subtract the same amount from both sides while keeping the equation balanced.
• Multiplication Property of Equality: We can multiply both sides of an equation by the same non-zero value, and the equation remains valid.
• Division Property of Equality: Like multiplication, dividing both sides by the same non-zero number maintains equality.

Understanding these properties helps in justifying each step when solving equations, ensuring we manipulate equations correctly while solving for a variable.

Equation Justification

Equation justification is all about explaining the reasons behind each transformation we perform on equations. It's like showing your work to prove you got the right answer.

For each step in solving the equation, we ask ourselves why a specific operation was necessary. This is where properties of equality come in—each operation must use one of these properties, ensuring the solution's validity.

For example, when we multiplied both sides of \(\frac{4}{5}x = 4x - 16\) by \(\frac{5}{4}\), we made this decision to eliminate the fraction and simplify the equation for easier manipulation. By explaining our reasoning, we give a clearer understanding of the process.

Multiplication Property of Equality

The multiplication property of equality is a powerful tool, especially when dealing with fractions. It allows us to clear any coefficient fractions by multiplying both sides of the equation by its reciprocal.

In our equation, we had \(\frac{4}{5}x = 4x - 16\). By multiplying both sides by \(\frac{5}{4}\), the left side becomes \(x\) because \(\frac{5}{4} \times \frac{4}{5} = 1\). This simplification helps focus on the core of the equation without distractions from fractions.

Always remember, whatever you do to one side, you must do to the other, which is the essence of the multiplication property in maintaining balance.

Subtraction Property of Equality

The subtraction property of equality allows us to subtract the same value from both sides of an equation. It's essentially used to gather like terms or isolate a variable.

In the problem, we rearranged \(x = 5x - 20\) to become \(x - 5x = 5x - 20 - 5x\). This step is crucial because by subtracting \(5x\) from both sides, we start simplifying the equation towards isolating x.

The key to using this property is to ensure you're reducing the complexity on both sides of the equation equally, gradually simplifying the equation.

Division Property of Equality

The division property of equality is often our last step in isolating a variable. By dividing both sides of the equation by the same non-zero number, we move towards our solution.

In our exercise, after simplifying to \(-4x = -20\), we used the division property to divide both sides by \(-4\). This isolates \(x\) and gives us \(x = 5\).

This property ensures the equation remains balanced, correcting any manipulated side so that their division results in the right solution. Always make sure the divisor is non-zero to avoid undefined expressions.