Problem 64 - Intermediate Algebra

Question

Solve each equation. Check each result. See Example 8. $$ 0.02 x+0.0175(15,000-x)=277.5 $$

Step-by-Step Solution

Answer

The solution is \(x = 6000\).

Step 1: Distribute the Second Term

Apply the distributive property to the term \(0.0175(15,000 - x)\). This means multiplying \(0.0175\) with both \(15,000\) and \(-x\). Therefore, the expression becomes:\[0.0175 imes 15,000 - 0.0175x = 262.5 - 0.0175x\].

Step 2: Combine Like Terms

Substitute the distributed terms from Step 1 into the original equation:\[0.02x + 262.5 - 0.0175x = 277.5\]Combine the \(x\) terms:\[(0.02x - 0.0175x) + 262.5 = 277.5\]This simplifies to:\[0.0025x + 262.5 = 277.5\].

Step 3: Isolate the Variable Term

Subtract \(262.5\) from both sides to isolate the term with \(x\):\[0.0025x = 277.5 - 262.5\]This simplifies to:\[0.0025x = 15\].

Step 4: Solve for x

To find \(x\), divide both sides by \(0.0025\):\[x = \frac{15}{0.0025}\]Calculate the division:\[x = 6000\].

Step 5: Check the Solution

Substitute \(x = 6000\) back into the original equation to verify the solution:\[0.02(6000) + 0.0175(15,000 - 6000) = 277.5\]Calculating the terms, we get:\[0.02 imes 6000 = 120\]\[15,000 - 6000 = 9000\]\[0.0175 imes 9000 = 157.5\]Adding these results, \(120 + 157.5 = 277.5\), verifying that \(x = 6000\) is correct.

Key Concepts

Distributive Property

In algebra, the distributive property is a fundamental rule that allows you to remove grouping symbols, such as parentheses, by distributing a factor across terms inside the parenthesis. This property is expressed as:

• For any numbers or expressions a, b, and c: \( a(b + c) = ab + ac \)

Let's break this down using our original equation: \(0.0175(15,000 - x)\). Here, you multiply \(0.0175\) by both \(15,000\) and \(-x\), resulting in \(0.0175 \times 15,000\) and \(-0.0175 \times x\), respectively. This transforms the expression into \(262.5 - 0.0175x\).
By distributing, you simplify the equation, making it easier to combine like terms in the next step.

Combining Like Terms

Combining like terms is a crucial step in simplifying algebraic equations. It involves adding or subtracting terms that have the same variable part. Here's how it's done:

• Identify like terms, which are terms with the same variable raised to the same power.
• Add or subtract the coefficients of these terms.

In our equation after applying the distributive property, we have: \(0.02x + 262.5 - 0.0175x = 277.5\).The terms \(0.02x\) and \(-0.0175x\) are like terms because they both involve the variable \(x\).
Combine them to simplify further: \((0.02x - 0.0175x) + 262.5 = 277.5\), resulting in \(0.0025x + 262.5 = 277.5\). This simplified equation sets the stage for isolating variables.

Isolating Variables

Isolating the variable is the process of manipulating an equation to get the variable by itself on one side of the equation, with everything else on the other side. This isolates the variable, making it possible to solve for its value. Here's how you do it:

• Use addition or subtraction to remove any constant terms from the side with the variable.
• Then, divide or multiply to solve for the variable term itself.

In our example, we simplify to have: \(0.0025x + 262.5 = 277.5\). Subtracting \(262.5\) from both sides gives us \(0.0025x = 15\).
Finally, divide each side by \(0.0025\) to isolate \(x\):
\(x = \frac{15}{0.0025}\). Solving this gives \(x = 6000\), thereby isolating the variable and determining its value.

Checking Solutions

Checking your solution verifies that it is correct and ensures that there were no mistakes during the calculation process. Here's how you do it:

• Take the solution you found and substitute it back into the original equation.
• Perform the calculations to ensure both sides of the equation are equal with the given solution.

For our solved equation, we substitute \(x = 6000\) back: \[0.02(6000) + 0.0175(15,000 - 6000) = 277.5\].Calculate each part separately:

• \(0.02 \times 6000 = 120\)
• Subtract \(15,000 - 6000 = 9000\)
• Then \(0.0175 \times 9000 = 157.5\)

Adding \(120\) and \(157.5\) results in \(277.5\), which matches the original equation.
This confirms that \(x = 6000\) is indeed the correct solution.