Problem 19 - Algebra for College Students

Question

Solve each equation. \(0.1 d+0.11(d+1500)=795\)

Step-by-Step Solution

Answer

The solution is \(d = 3000\).

Step 1: Distribute the 0.11 inside the parentheses

Start by distributing the 0.11 to both terms inside the parentheses. This gives us the equation: \[ 0.1d + 0.11d + 165 = 795 \]

Step 2: Combine like terms

Combine the terms that contain \(d\) on the left side of the equation. This simplifies to:\[ 0.21d + 165 = 795 \]

Step 3: Subtract 165 from both sides

To isolate the term with \(d\), subtract 165 from both sides:\[ 0.21d = 795 - 165 \]This simplifies to:\[ 0.21d = 630 \]

Step 4: Divide both sides by 0.21

To solve for \(d\), divide both sides of the equation by 0.21:\[ d = \frac{630}{0.21} \]Calculating this gives:\[ d = 3000 \]

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra used to simplify expressions. It involves multiplying a single term by each term inside a set of parentheses. This property is vital because it allows you to break down complex expressions into simpler parts.
In our example, the equation given is:

• 0.1d + 0.11(d + 1500) = 795

The distributive property tells us to multiply 0.11 by each term inside the parentheses, giving:

• 0.1d + 0.11d + 165 = 795

The operation makes the expression easier to handle by eliminating the parentheses. Essentially, you distribute the 0.11 over both the variables and the constant within the parentheses, helping to simplify the equation for the next steps.

Combining Like Terms

Combining like terms is the process of simplifying algebraic expressions by merging terms that have the same variable. This makes the equation more manageable and is crucial for solving equations.
In the equation:

• 0.1d + 0.11d + 165 = 795

The terms 0.1d and 0.11d are like terms because they both contain the variable 'd'. We combine them by adding the coefficients:

• 0.1 + 0.11 = 0.21

So the equation simplifies to:

• 0.21d + 165 = 795

This process helps in clearing out extra terms, making it easier to focus on the remaining parts of the expression or equation.

Isolating Variables

Isolating variables is a crucial step when solving equations. It involves getting the variable on one side of the equation by itself. This step is necessary to find the value of the unknown in any algebraic equation.
To isolate the variable 'd' in our equation 0.21d + 165 = 795, we first need to remove the constant term 165 from the left side by subtracting 165 from both sides.

• This gives us: 0.21d = 795 - 165

Simplify the right side:

• 0.21d = 630

By doing this, 'd' is alone with its coefficient on the left side, setting it up perfectly for the next step, which is to solve for 'd' by division.

Division in Algebra

Division in algebra is the process of solving for a variable once it’s isolated with a coefficient. This involves dividing both sides of the equation by the coefficient of the variable.
In our equation 0.21d = 630, the next step is to divide both sides by 0.21 to solve for 'd':

• d = \( \frac{630}{0.21} \)

When you calculate this, you get:

• d = 3000

Dividing both sides by the coefficient effectively cancels the coefficient out on one side, leaving the variable by itself. Remember that whatever operation you perform on one side of the equation, you must also perform on the other to maintain balance; this is a fundamental rule in algebra.