Problem 28 - Introductory Algebra for College Students

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 \(x - \frac{3}{5} = \frac{7}{10}\) is \(x = \frac{13}{10}\).

Step 1: Apply Addition Property of Equality

Add \(\frac{3}{5}\) to both sides of the equation \(x - \frac{3}{5} = \frac{7}{10}\) to rid the left side of the equation of the fraction. This gives us \(x - \frac{3}{5} + \frac{3}{5} = \frac{7}{10} + \frac{3}{5}\), which simplifies to \(x = \frac{7}{10} + \frac{3}{5}\).

Step 2: Simplify the Right Side

To simplify \(\frac{7}{10} + \frac{3}{5}\), first convert the fraction \(\frac{3}{5}\) to have a denominator of 10, becoming \(\frac{6}{10}\). The equation now becomes \(x = \frac{7}{10} + \frac{6}{10}\).

Step 3: Add the Fractions

Add the fractions on the right side of the equation. This gives us \(x = \frac{13}{10}\).

Step 4: Verification

Substitute \(x = \frac{13}{10}\) back into the original equation \(x-\frac{3}{5}=\frac{7}{10}\) to verify that it is the correct solution. This results in \(\frac{13}{10} - \frac{3}{5} = \frac{13}{10} - \frac{6}{10} = \frac{7}{10}\), which verifies the solution.

Key Concepts

Solving Equations

Solving equations is a fundamental concept in mathematics involving finding values for variables that make the equation true. In the equation \(x - \frac{3}{5} = \frac{7}{10}\), we want to find the value of \(x\) that satisfies both sides of the equation. The addition property of equality is a key tool in solving such equations. By adding the same number to both sides of an equation, the two sides remain equal. To apply this property, start by adding \(\frac{3}{5}\) to both sides to isolate \(x\) on one side. This operation ensures that \(x - \frac{3}{5} + \frac{3}{5} = \frac{7}{10} + \frac{3}{5}\), simplifying to \(x = \frac{7}{10} + \frac{3}{5}\). Once \(x\) is isolated, the solution process involves simplifying and finding \(x\)'s value. Utilizing these steps consistently will help solve equations effectively.

Fractions

Fractions represent parts of a whole and are significant in equations when not all values are whole numbers. Understanding how to manipulate fractions is crucial when solving equations that contain them. In the given equation, both sides include fractions \(\frac{3}{5}\) and \(\frac{7}{10}\). Before adding these fractions, they need a common denominator. To find a common denominator, such as 10, convert \(\frac{3}{5}\) to \(\frac{6}{10}\). Now both fractions share the same base, making it easier to add them: \(\frac{7}{10} + \frac{6}{10}\). When the numerators are added, the fraction becomes \(\frac{13}{10}\). Understanding fractions and their behaviors ensure accurate results, particularly when adding, subtracting, and simplifying them in equations. Keep practicing these skills to enhance your confidence when dealing with fractions.

Verification of Solutions

Verifying solutions means checking if your proposed answer satisfies the original equation. It's essential to ensure that no mistakes were made during the calculations. After determining \(x = \frac{13}{10}\), go back to the initial equation \(x - \frac{3}{5} = \frac{7}{10}\) and plug this value of \(x\) into it. Substitute and simplify, \(\frac{13}{10} - \frac{3}{5}\), by first using the common denominator approach just as before, converting \(\frac{3}{5}\) into \(\frac{6}{10}\). This simplification results in \(\frac{13}{10} - \frac{6}{10} = \frac{7}{10}\), which confirms that our solution is correct. By consistently verifying solutions, you can be confident in your math work and avoid errors.