Problem 58 - Elementary Algebra

Question

Solve each of the following equations and express the solutions in decimal form. Your calculator might be of some help. $$9 x-0.37=0.35$$

Step-by-Step Solution

Answer

The solution is \(x = 0.08\).

Step 1: Isolate the Variable Term

Begin by moving the constant term to the right side of the equation. You have the equation: \(9x - 0.37 = 0.35\).Add \(0.37\) to both sides to remove it from the left side:\(9x - 0.37 + 0.37 = 0.35 + 0.37\).This simplifies to:\(9x = 0.72\).

Step 2: Solve for the Variable

Now that you've isolated the term with \(x\), divide both sides of the equation by \(9\) to solve for \(x\).\(x = \frac{0.72}{9}\).Use a calculator to divide:\(x = 0.08\).

Key Concepts

Isolating the Variable

When solving linear equations, isolating the variable is a crucial step. It means getting the variable (like \(x\) in our example) by itself on one side of the equation. This helps you see the value of the variable more clearly. Let's break it down:

• Start with your equation: \(9x - 0.37 = 0.35\).
• Notice the \(-0.37\) is combined with \(9x\). You need to clear this by doing the opposite operation.
• Add \(0.37\) to both sides, which cancels it on the left side: \(9x - 0.37 + 0.37 = 0.35 + 0.37\).
• This yields \(9x = 0.72\), effectively isolating \(9x\).

Remember, isolating involves moving all other terms to the opposite side of the equation. Aim for a clean equation such as \(cx = d\), where \(c\) and \(d\) are constants, with \(x\) standing alone on one side.

Decimal Solutions

Decimal solutions are expressed in decimal form rather than fractions. They are often easier to interpret, especially when precise values are needed. Here's why decimals are commonly used:

• Decimals can provide a clear and precise answer when dealing with real-world problems.
• They are especially useful when using calculators, which usually return results in decimal form.
• In our example, \(x = 0.08\) is a straightforward decimal, indicating a fraction value of \(\frac{8}{100}\), or more simply, \(\frac{2}{25}\).

When converting a division result into a decimal, ensure you use a calculator if needed. This precision can be crucial for maintaining accuracy in calculations, especially in scientific or financial contexts.

Solving Equations

Solving equations involves finding the value of the variable that makes the equation true. This process consists of a few systematic steps. Let's review them:

• First, you aim to simplify both sides of the equation as much as possible.
• In our exercise, after isolating \(9x\), we ended up with \(9x = 0.72\).
• The next move is to divide both sides of the equation by the coefficient of \(x\), which is \(9\) in this case.
• Your equation: \(x = \frac{0.72}{9}\).
• Utilize a calculator to perform this division accurately, resulting in \(x = 0.08\).

These steps ensure that the variable is completely isolated, and the solution is in the desired form. Solving equations systematically builds strong mathematical foundations, allowing you to tackle more complex problems with ease.