Problem 26 - 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. $$t+\frac{2}{3}=-\frac{7}{6}$$

Step-by-Step Solution

Answer

The solution to the equation is \(t = -\frac{11}{6}\)

Step 1: Solve for t

Subtract \(\frac{2}{3}\) from both sides of the equation using the addition property of equality. This results in: \(t = -\frac{7}{6} - \frac{2}{3}\)

Step 2: Simplify Right Side

Simplify the right side of the equation by performing the subtraction. A common denominator of 6 can be used for fractions. The equation becomes: \(t = -\frac{7}{6} - \frac{4}{6}\)

Step 3: Further Simplify Equation

When you perform the subtraction on the right-hand side, you get: \(t = -\frac{11}{6}\)

Step 4: Check Solution

Substitute \(t = -\frac{11}{6}\) back into the original equation to verify that it satisfies the equation: \(-\frac{11}{6} + \frac{2}{3} = -\frac{7}{6} .\) Simplifying the left side gives \(-\frac{7}{6}\), which equals the right side of the equation. Thus our solution is correct.

Key Concepts

Addition Property of Equality

Understanding the addition property of equality is crucial for solving linear equations effectively. In essence, it states that you can add or subtract the same number from both sides of an equation without changing the solution. This property maintains the balance of the equation, much like a scale in equilibrium.

For example, if you have an equation like \( x + 2 = 5 \), applying this property allows you to subtract 2 from both sides, resulting in \( x = 3 \). It's the cornerstone of isolating variables and moving terms from one side of an equation to the other. Remember, whatever operation you perform on one side, you must do the same to the other to keep the equality true.

Simplify Fractions

Simplifying fractions is a fundamental skill when dealing with equations involving fractions. The key is to express the fractions with the smallest numerator and denominator possible. This often requires finding a common denominator, which makes adding, subtracting, multiplying, or dividing fractions easier.

Take the example where you have \( \frac{2}{5} \) and \( \frac{3}{5} \), your common denominator is 5, and combining them is straightforward. However, if you have fractions with different denominators, like \( \frac{2}{3} \) and \( \frac{7}{6} \), find a shared multiple that can act as your new common denominator. For these two fractions, 6 works well as you only need to adjust the fraction with the denominator of 3.

Equation Solving Steps

The step-by-step process to solve an equation can vary slightly depending on the equation's complexity. Generally, though, the following steps are a solid guideline:

• Identify the variable you need to solve for and the operations involved.
• Use the addition property of equality to eliminate any additions or subtractions from both sides.
• Isolate the variable by using inverse operations, such as subtracting or adding to both sides or multiplying or dividing both sides by the same number.
• Simplify each side as much as possible by combining like terms and reducing fractions.
• Verify your solution by substituting it back into the original equation.

These steps, applied consistently, will guide you to the correct solution for most linear equations.

Checking Equations Solutions

After solving an equation, it's important not to skip the step of checking your solution. This verification process ensures that the value you found for the variable indeed makes the original equation true. Substitution is the key to this process. Take the proposed solution, and plug it back into the original equation in place of the variable.

Ensure that you simplify any arithmetic on both sides of the equation. If both sides reduce to the same value, your solution is correct. If not, you may need to retrace your steps to find any errors. This careful check not only confirms your solution but also deepens your understanding of the equation's behavior.