Problem 20 - Algebra for College Students

Question

Solve each equation. \(0.8 x+0.9(850-x)=715\)

Step-by-Step Solution

Answer

The solution is \(x = 500\).

Step 1: Distribute Terms

Begin by distributing the 0.9 across the expression inside the parentheses. This gives the equation: \[0.8x + 0.9 \cdot 850 - 0.9x = 715\] which simplifies to: \[0.8x + 765 - 0.9x = 715\]

Step 2: Combine Like Terms

Next, combine the \(x\)-terms on the left side of the equation. Subtract 0.9x from 0.8x to get:\[-0.1x + 765 = 715\]

Step 3: Isolate the Variable Term

To isolate the variable term, subtract 765 from both sides of the equation: \[-0.1x = 715 - 765\] which simplifies to: \[-0.1x = -50\]

Step 4: Solve for the Variable

Finally, solve for \(x\) by dividing both sides by -0.1: \[x = \frac{-50}{-0.1}\] which simplifies to: \[x = 500\]

Key Concepts

Step-by-Step Solution

A step-by-step solution is like a guided tour through a math problem, taking each part one by one. It breaks down complex problems into smaller, manageable steps. This method not only helps in understanding but also builds confidence in solving similar problems independently.

Let's look at an example with the equation: \(0.8x + 0.9(850-x) = 715\). Step-by-step solutions begin by addressing each component separately, ensuring clarity. Following this structured approach is essential to achieve the correct solution. By methodically distributing, combining like terms, isolating the variable, and solving for \(x\), each step brings us closer to the final answer. By the end of the process, you not only solve the problem but also enhance your problem-solving skills.

Combining Like Terms

Combining like terms is an important concept in simplifying algebraic expressions. It involves grouping and simplifying terms in an equation that have the same variable part. This is crucial as it transforms a complicated expression into something more straightforward.

• In the equation \(0.8x + 0.9(850-x) = 715\), after distributing, we obtain \(0.8x + 765 - 0.9x = 715\).
• The like terms here are \(0.8x\) and \(-0.9x\), which can be combined to get \(-0.1x\).

This simplification helps in reducing the equation into a form where solving becomes easier. Remember, the coefficients of the like terms should be combined while retaining the common variable and keeping the equation balanced.

Isolating the Variable

Isolating the variable means getting the variable by itself on one side of the equation. This is an essential step because it allows us to solve for the variable directly.

In our example, after reaching \(-0.1x + 765 = 715\), we isolate \(x\) by removing any added or subtracted terms from the side with \(x\). Subtract 765 from both sides to get \(-0.1x = 715 - 765\), which simplifies to \(-0.1x = -50\).

• This step helps to transition the equation to a stage where you can directly solve for \(x\).
• It's important to perform the same operation on both sides to maintain the equality.

By having the variable isolated, you make the next steps much clearer and easier to handle.

Distributing Terms

Distributing terms is all about applying the distributive property of multiplication over addition or subtraction, which is an essential tool in algebra.

In the original equation \(0.8x + 0.9(850-x) = 715\), the part \(0.9(850-x)\) requires distribution. This means multiplying \(0.9\) with both \(850\) and \(-x\).

• After distribution, the equation becomes \(0.8x + 765 - 0.9x = 715\).
• Distributing helps simplify complex expressions, making them easier to solve.

This concept is useful in breaking down expressions into simpler parts, allowing us to effectively proceed with solving equations. Understanding how to distribute properly is crucial as it sets the foundation for the next steps in solving linear equations.