Problem 61 - Algebra for College Students

Question

Solve each equation and express the solutions in decimal form. Be sure to check your solutions. Use your calculator whenever it seems helpful. \(0.6(d-4.8)=7.38\)

Step-by-Step Solution

Answer

The solution is \(d = 17.1\).

Step 1: Distribute the Coefficient

We start by distributing the coefficient of 0.6 to both terms inside the parentheses. The equation is:\[ 0.6(d-4.8) = 7.38 \]After distributing, it becomes:\[ 0.6d - 0.6 \times 4.8 = 7.38 \] Calculate \(0.6 \times 4.8\):\[ 0.6 \times 4.8 = 2.88 \] Therefore, the equation simplifies to:\[ 0.6d - 2.88 = 7.38 \]

Step 2: Isolate the Variable Term

Next, we need to isolate the term with the variable \(d\). To do that, add 2.88 to both sides of the equation:\[ 0.6d - 2.88 + 2.88 = 7.38 + 2.88 \] The equation becomes:\[ 0.6d = 10.26 \]

Step 3: Solve for the Variable

Now, divide both sides of the equation by 0.6 to solve for \(d\):\[ d = \frac{10.26}{0.6} \] Using a calculator, calculate \(\frac{10.26}{0.6}\):\[ \frac{10.26}{0.6} = 17.1 \] Thus, \(d = 17.1\).

Step 4: Check the Solution

To ensure the solution is correct, substitute \(d = 17.1\) back into the original equation.Start with:\[ 0.6(d-4.8) = 7.38 \] Substitute \(d = 17.1\):\[ 0.6(17.1 - 4.8) \] Calculate inside the parentheses:\[ 17.1 - 4.8 = 12.3 \] Now, calculate the left side of the equation:\[ 0.6 \times 12.3 = 7.38 \] Since both sides of the equation match, the solution is verified.

Key Concepts

Distributive Property

Understanding how to use the distributive property effectively can simplify many problems, including those involving linear equations. In this exercise, we started with the equation \( 0.6(d-4.8) = 7.38 \). The distributive property allows us to multiply a single term by each term inside the parentheses.

Here's how it works:

• Multiply \(0.6\) by \(d\), which gives \(0.6d\).
• Next, multiply \(0.6\) by \(-4.8\). This results in \(-2.88\).

The revised equation after applying the distributive property is \(0.6d - 2.88 = 7.38\).

Using the distributive property simplifies the expression. It is an invaluable tool for tackling equations where terms need to be expanded for further manipulation.

Variable Isolation

Isolating the variable is a crucial step toward solving equations. It involves rearranging the equation so that the variable (in this case, \(d\)) is by itself on one side. After applying the distributive property, the equation was \(0.6d - 2.88 = 7.38\).

To isolate \(d\):

• Add 2.88 to both sides, which cancels the \(-2.88\) on the left.
• This yields \(0.6d = 10.26\).

Remember, whatever you do to one side of the equation, you must do to the other. This maintains balance and equality in the equation. Once isolated, the coefficient of the variable can be dealt with in the next steps.

Decimal Solutions

In this exercise, the solution involves calculations that lead to decimals. Decimal solutions are common, especially when dealing with realistic scenarios or precise measurements.

After isolating the variable \(d\), we found ourselves needing to solve \(0.6d = 10.26\) by dividing both sides by \(0.6\). This yields \(d = 17.1\), which is our decimal solution.

Working with decimals requires careful attention to accuracy, especially during division or multiplication. Using a calculator, as detailed in the next section, can help ensure precision in your calculations.

Calculator Usage

Calculators are essential tools, especially when dealing with complex arithmetic or verifying solutions. In our exercise, a calculator played a critical role in several steps.

To solve \(d = \frac{10.26}{0.6}\), handy calculator usage helps achieve an accurate result of \(17.1\). Calculators also help when verifying solutions by redoing operations or checking for consistency, like ensuring that \(0.6 \times 12.3 = 7.38\), which confirms our solution.

While calculators handle the arithmetic, understanding the process ensures you're using the tool effectively. Check your work and understand each step, using the calculator as a reliable aid, not a crutch.