Problem 24 - Algebra for College Students

Question

Solve each equation. \(0.09 x=1650-0.12(x+5000)\)

Step-by-Step Solution

Answer

The solution is \(x = 5000\).

Step 1: Distribute and Simplify

First, let's distribute the 0.12 across the terms inside the parentheses on the right-hand side of the equation. This gives us:\[0.09x = 1650 - 0.12x - 0.12 imes 5000\]Calculate \(0.12 \times 5000\) which is 600. Substitute this back into the equation:\[0.09x = 1650 - 0.12x - 600\]Now simplify the right-hand side:\[0.09x = 1050 - 0.12x\]

Step 2: Collect Like Terms

To isolate \(x\), we'll bring all terms involving \(x\) to one side of the equation. Add \(0.12x\) to both sides:\[0.09x + 0.12x = 1050\]Combine the \(x\) terms:\[0.21x = 1050\]

Step 3: Solve for \(x\)

We need to solve for \(x\) by dividing both sides of the equation by 0.21:\[x = \frac{1050}{0.21}\]Calculate \(\frac{1050}{0.21}\) to get the value of \(x\):\[x = 5000\]

Key Concepts

Distributive Property

The Distributive Property is a fundamental tool in algebra that helps us simplify equations by distributing a single term across terms inside a parenthesis. For example, consider an expression where a number is multiplied by a sum or difference inside the parenthesis, like in our exercise's original equation:

• 0.12(x + 5000)

The distributive property allows us to multiply the 0.12 with each term inside the parenthesis:

• 0.12 \( \times \) x
• 0.12 \( \times \) 5000

After distributing the terms, the expression becomes:

• 0.12x + 600

This step lays the groundwork for simplifying the equation further, making the problem easier to solve. By using the Distributive Property, we can break down complex problems into manageable pieces. Remember this handy property whenever you encounter terms inside parentheses.

Collecting Like Terms

Once we have simplified an equation using the Distributive Property, the next step is to collect like terms. Like terms are terms that have the same variable raised to the same power. In our example:

• Original Step: \(0.09x = 1650 - 0.12x - 600\)
• Simplified Step: \(0.09x = 1050 - 0.12x\)

Here, both sides of the equation have terms involving the variable \(x\):

• 0.09x on the left
• -0.12x on the right

To solve for \(x\), we move all terms involving \(x\) to one side of the equation. This means adding 0.12x to both sides, which combines the \(x\) terms:

• 0.09x + 0.12x = 1050
• 0.21x = 1050

By combining like terms, we simplify the equation to a point where it becomes easier to isolate and solve for \(x\). Collecting like terms is an essential skill in algebraic manipulation, streamlining equations for efficient solving.

Solving Equations

After simplifying the equation by using the Distributive Property and collecting like terms, the final step is solving the equation. Our equation after combining like terms is:

• 0.21x = 1050

To find the value of \(x\), we need to isolate it. This is done by performing the same operation on both sides of the equation:

• Divide both sides by 0.21

This gives:

• x = \(\frac{1050}{0.21}\)

Carrying out the division yields:

• x = 5000

Thus, the solution to the equation is \(x = 5000\). Solving equations involves isolating the variable and performing operations to both sides of the equation to maintain equality. Always check your work by plugging the solution back into the original equation to ensure accuracy.