Problem 38 - Algebra 1

Question

Solve the equation. $$0=\frac{4}{5} d$$

Step-by-Step Solution

Answer

The solution to the given equation \(0 = \frac{4}{5}d\) is \(d=0\).

Step 1: Understand the Equation

The given equation is \(0 = \frac{4}{5}d\). The goal is to find the value of \(d\). In other words, we are asked to solve for \(d\).

Step 2: Isolate the Variable \(d\)

In order to isolate \(d\), we need to get rid of the coefficient \(\frac{4}{5}\). But since it is being multiplied by \(d\), you could think of dividing both sides by \(\frac{4}{5}\) to cancel it out. However, since we know that any number divided by itself is 1 and any number multiplied by zero is zero. Therefore, \(d\) is zero.

Step 3: Final Solution

Now, the variable \(d\) is isolated and has been found to be zero, which is the solution to the given equation.

Key Concepts

Isolating Variables

Isolating variables is a crucial step in solving equations. Here, it means making the unknown variable appear all by itself on one side of the equation. To isolate a variable, we often need to remove other terms or coefficients that are "attached" to the variable.

In our exercise, we want to isolate the variable \(d\) in the equation \(0 = \frac{4}{5}d\). Initially, \(d\) is multiplied by \(\frac{4}{5}\). To get \(d\) by itself, we need to remove this fraction:

• Since \(\frac{4}{5}\) is a fraction, we can divide both sides of the equation by \(\frac{4}{5}\) to cancel it out of the right side.
• Alternatively, you can multiply both sides by the reciprocal of \(\frac{4}{5}\), which is \(\frac{5}{4}\).

By using either method, we eliminate the fraction, thus isolating \(d\). However, in our specific equation, we notice it's set equal to zero. Knowing that zero divided by or multiplied by any non-zero number remains zero simplifies the process a lot. This means the solution becomes obvious – \(d\) itself equals zero once isolated.

Multiplication and Division in Equations

Multiplication and division help manipulate equations so that we can isolate and solve for variables. If a variable is being multiplied by a number, as in our equation, we can divide by that number to solve for the variable.

In the equation \(0 = \frac{4}{5}d\):

• The variable \(d\) is multiplied by the fraction \(\frac{4}{5}\).
• To "undo" this multiplication and solve for \(d\), we can divide both sides by \(\frac{4}{5}\).

By performing this operation equally on both sides, not only do we respect the balance of the equation, but it also allows us to clear the coefficient from \(d\). In practice, especially with zero, division simplifies the process further because dividing or multiplying zero by any constant almost always results in zero. Thus, \(d = 0\), as you complete the division step.

Understanding Coefficients

Coefficients are the numbers that multiply the variables in an equation. They provide a scaling factor for the variable they are attached to. Grasping their role is important because they tell us how much of the variable is being considered in the equation.

In the given equation \(0 = \frac{4}{5}d\), \(\frac{4}{5}\) is the coefficient of \(d\). Understanding what a coefficient does helps in knowing why we need to eliminate it to solve the variable:

• The coefficient indicates that \(d\) is being multiplied by \(\frac{4}{5}\).
• When solving equations, to find \(d\)'s standalone value, we need to "cancel" this effect.
• This is done by executing an operation that neutralizes the coefficient, like division by the coefficient.

Recognizing that zero times or divided by any coefficient remains zero is essential in simplified scenarios like this, showing us instantly why \(d = 0\). Coefficients thus shape the initial form of the equation and guide how we manipulate the terms until the variable is isolated.