Problem 38 - Introductory Algebra for College Students

Question

Solve each equation using the addition property of equality. Be sure to check your proposed solutions. $$\frac{7}{3}=-\frac{5}{2}+z$$

Step-by-Step Solution

Answer

The solution to the equation is \( z = \frac{29}{6} \)

Step 1: Rewrite the equation for clarity

First, let's rewrite the equation to make it clearer.\[ \frac{7}{3} = -\frac{5}{2} + z \]

Step 2: Use the Addition Property of Equality to isolate z

To isolate the z term, we'll use the rule of equality which states that we can add the same amount to both sides of an equation without changing its balance. So, add \( \frac{5}{2} \) to both sides of the equation.\[ \frac{7}{3} + \frac{5}{2} = z \]

Step 3: Simplify the equation to find the value of z

Then, calculate the addition on the left side. Make sure both fractions have the same denominator.\[ z = \frac{14}{6} + \frac{15}{6} = \frac{29}{6}\]

Step 4: Check the solution

Finally, substitute \( z = \frac{29}{6} \) into the original equation to confirm the correct solution. The left-hand side becomes \( \frac{7}{3} \), and the right-hand side becomes \( -\frac{5}{2} + \frac{29}{6} = \frac{7}{3} \), so the equation is valid.

Key Concepts

Understanding Fractions

Fractions represent parts of a whole and are composed of two main parts: a numerator (the top number) and a denominator (the bottom number). When dealing with fractions, understanding how to find a common denominator is crucial, especially for addition and subtraction.

• To add fractions, ensure they have the same denominator.
• Multiply the numerators and denominators accordingly to convert them.
• Once converted, add the numerators and keep the denominator the same.

In the example given, the fractions \( \frac{7}{3} \) and \( \frac{5}{2} \) needed a common denominator. Both 6 is the least common multiple of 3 and 2, converting them to \( \frac{14}{6} \) and \( \frac{15}{6} \) respectively.

Solving Equations

Solving equations means finding what value the variable represents in order to make the equation true. Equations often involve operations like addition, subtraction, multiplication, and division.

• Start by simplifying each side of the equation if needed.
• Use balance methods to keep the equation equal and solve for the variable.
• Check your solution by plugging it back into the original equation.

In our exercise, the equation is centered around isolating the variable \( z \). By adding \( \frac{5}{2} \) to both sides, we maintain equality, allowing us to solve for \( z \). The result, \( z = \frac{29}{6} \), checks out when substituted back, verifying our solution.

The Substitution Method

The substitution method involves replacing a variable with a known value to check if an equation holds true. It's a validation step for your solution.

• Identify the solution for the variable from the solved equation.
• Insert the known solution back into the original equation.
• Check if both sides of the equation balance.

Once we solved for \( z \) and found \( z = \frac{29}{6} \), we substituted it back into the original equation. Replacing \( z \) with \( \frac{29}{6} \) gives both sides an equal value of \( \frac{7}{3} \), confirming our solution is accurate.