Problem 28 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve each equation. Using the addition property of equality. Be sure to check your proposed solutions. $$x-\frac{3}{5}=\frac{7}{10}$$

Step-by-Step Solution

Answer

The solution to the equation is \(x =\frac{13}{10}\) or \(x = 1.3\).

Step 1: Isolate x

To isolate x, add \(\frac{3}{5}\) to both sides of the equation. This gives us: \(x = \frac{7}{10} + \frac{3}{5}\).

Step 2: Add the Fractions

To add the fractions \(\frac{7}{10}\) and \(\frac{3}{5}\), we need to find a common denominator. The least common multiple of 10 and 5 is 10, so convert \(\frac{3}{5}\) to \(\frac{6}{10}\). Now the equation is: \(x = \frac{7}{10} + \frac{6}{10}\)

Step 3: Simplify the Right-Hand Side

Now add the fractions \(\frac{7}{10} + \frac{6}{10}\). This gives us \(x = \frac{13}{10}\), or we could say that \(x = 1.3\) if we convert the fraction to a decimal.

Step 4: Check the Proposed Solution

To check the solution, substitute \(x = \frac{13}{10}\) in the original equation and see if both sides of the equation are equal. \(\frac{13}{10} - \frac{3}{5} = \frac{7}{10}\) simplifies to \(\frac{7}{10} = \frac{7}{10}\), which is true. Hence the solution is correct.

Key Concepts

Addition Property of Equality

When we talk about solving equations, the addition property of equality is a fundamental concept. This property states that you can add the same number to both sides of an equation without changing its truth. It's like keeping the equation's balance. For example, if you have \( x - \frac{3}{5} = \frac{7}{10} \) and you want to isolate \( x \), you need to get rid of the \(-\frac{3}{5}\). To do this, add \(\frac{3}{5}\) to both sides. This transforms the equation to \( x = \frac{7}{10} + \frac{3}{5} \). By maintaining the balance through addition, we're shaping the equation to solve for \( x \).

Common Denominator

When you need to add or subtract fractions, having a common denominator is crucial. It refers to making the denominators in fractions the same number. Why do we need this? Because it allows us to easily combine fractions, much like adding apples and apples rather than apples and oranges! In the equation \( x = \frac{7}{10} + \frac{3}{5} \), the fractions \( \frac{7}{10} \) and \( \frac{3}{5} \) need the same base to be added. Since 10 is the least common multiple of both denominators 10 and 5, we convert \( \frac{3}{5} \) into \( \frac{6}{10} \). Now with the common denominator, you can comfortably add them!

Fraction Addition

Once fractions share the same denominator, they're ready to be added. Think of it like combining like terms. For instance, with \( \frac{7}{10} + \frac{6}{10} \), you can add the numerators together while keeping the denominator the same. This gives \( \frac{13}{10} \), a straightforward and clear answer. Simplifying is often an option as well, so this result of \( \frac{13}{10} \) can further be expressed as \( 1.3 \). Remember:

• Convert fractions to have a common denominator.
• Add the numerators.
• Keep the denominator constant.

Solution Verification

Now, to verify our solution, it's crucial to ensure our proposed value of \( x \) satisfies the original equation. Inserting \( x = \frac{13}{10} \) back into \( x - \frac{3}{5} = \frac{7}{10} \), we perform the calculation to check both sides match. Substitute the \( x \) value:

• \(\frac{13}{10} - \frac{3}{5} = \frac{7}{10}\). Convert \(\frac{3}{5}\) to \(\frac{6}{10}\).
• Perform the fraction subtraction \(\frac{13}{10} - \frac{6}{10} = \frac{7}{10}\).

Since the equation holds true — \(\frac{7}{10} = \frac{7}{10}\), our solution checks out. This ensures accuracy and validates our steps are correct!