Problem 23 - Prealgebra

Question

Solve each equation. $$0.08 x+0.09(x+2000)=690$$

Step-by-Step Solution

Answer

The solution is \(x = 3000\).

Step 1: Distribute the Coefficient

Begin by distributing the 0.09 to both terms inside the parentheses: \(0.09(x+2000) = 0.09x + 0.09 \times 2000\). This simplifies to \(0.09x + 180\).

Step 2: Combine Like Terms

Rewrite the equation by combining like terms: \(0.08x + 0.09x + 180 = 690\). Add \(0.08x\) and \(0.09x\) to get \(0.17x\). The equation becomes \(0.17x + 180 = 690\).

Step 3: Isolate the Variable Term

Subtract 180 from both sides to get all terms involving \(x\) on one side: \(0.17x + 180 - 180 = 690 - 180\). This simplifies to \(0.17x = 510\).

Step 4: Solve for x

Divide both sides by 0.17 to solve for \(x\): \(x = \frac{510}{0.17}\). Calculating this gives \(x = 3000\).

Key Concepts

Distributive Property

The distributive property is a cornerstone of algebra that allows us to simplify expressions by distributing a single term over multiple terms inside parentheses. This means that if you have an expression like:

• \( a(b + c) \)

You can distribute \( a \) to both \( b \) and \( c \), which translates to:

• \( ab + ac \)

In our exercise, we used this property on \( 0.09(x + 2000) \), which gave us the result:

• \( 0.09x + 180 \)

The key here is to multiply every term inside the parentheses by the term outside, making equations simpler to manage.
Once you master this, handling complex equations becomes much easier.

Combining Like Terms

Combining like terms is an essential process in solving linear equations. This simply refers to the task of summing all terms in an equation that have the same variable part.
In our solution, we had the equation:

• \(0.08x + 0.09x + 180 = 690\)

The terms \(0.08x\) and \(0.09x\) are considered 'like terms' because they both involve the variable \(x\). By adding them together, we simplified it to:

• \(0.17x + 180 = 690\)

This makes the equation simpler and easier to solve.
While combining like terms, always ensure all coefficients of the same variables are summed up. This helps reduce the complexity of your equations, allowing for more straightforward solving.

Isolate the Variable

Isolating the variable is a crucial step to finding the solution to an equation. Once you've combined like terms, the next task is to get the variable on one side of the equation by itself.
In our example, the equation was:

• \(0.17x + 180 = 690\)

Here, we subtracted 180 from both sides to maintain equality and "isolate" \(x\). This resulted in:

• \(0.17x = 510\)

By having \(x\) on its own, it’s much easier to compute its value in the next steps.
Isolating variables is about reverse-engineering the operations applied to the variable, making sure you're one step closer to getting the solution.

Step-by-Step Math Solutions

Breaking down complex problems into smaller, manageable parts allows us to tackle them effectively. Each step in solving linear equations builds on the last, ensuring students understand the rationale behind every move.
Here's what was done in the context of our problem:

• Started with the **distributive property** to open up the parentheses.
• Moved on to **combine like terms** to simplify the expression.
• Worked to **isolate the variable** by rearranging the equation.
• Finally, calculated the solution for **\(x\)**.

This method not only produces the right answer but also builds comprehension, explaining why each action is taken. Always strive to follow a logical progression of steps to deepen understanding and solve equations effectively.