Problem 23 - Introductory Algebra for College Students

Question

Solve each equation using the addition property of equality. Be sure to check your proposed solutions. $$x+\frac{1}{3}=\frac{7}{3}$$

Step-by-Step Solution

Answer

The solution is \(x=2\)

Step 1: Set up the Equation

Initially, we have the equation \(x+\frac{1}{3}=\frac{7}{3}\). Our aim is to find the value of \(x\).

Step 2: Applying the Addition property of equality

The addition property of equality states that adding the same value to both sides of an equation yields an equivalent equation. To isolate \(x\) on the left-hand side we subtract \(\frac{1}{3}\) from both sides to cancel out the \(\frac{1}{3}\) on the left side. So, \(x+\frac{1}{3} - \frac{1}{3} = \frac{7}{3} - \frac{1}{3}\) which simplifies to \(x = \frac{6}{3}\).

Step 3: Reduce to lowest terms

\(x = \frac{6}{3}\) can further be simplified to \(x = 2\). Therefore the solution to the equation is \(x = 2\). We've isolated \(x\) and this is our much-anticipated solution.

Step 4: Check the solution

To ensure that \(x = 2\) is the correct solution, we substitute \(x\) with 2 in the original equation. This yields \(2 + \frac{1}{3} = \frac{7}{3}\) which simplifies to \(\frac{7}{3} = \frac{7}{3}\). Since both sides of the equation are equal, we can confirm that our solution is correct.

Key Concepts

Solving Equations

When it comes to solving equations, the main goal is to find the value that makes an equation true. In mathematical terms, we are looking for the value of the variable, in this case, the value of \(x\).

To solve an equation, we often rely on properties like the addition property of equality. This property allows us to manipulate the equation to isolate the variable on one side. When both sides of an equation are balanced, whatever we add or subtract from one side, we must do to the other to maintain that balance.

In the exercise given, we started with the equation \(x+\frac{1}{3}=\frac{7}{3}\). Our task was to solve for \(x\). By subtracting \(\frac{1}{3}\) from both sides, we effectively "moved" it to the other side. This left us with \(x=\frac{6}{3}\), which we simplified to \(x=2\). This method of isolating the variable by using the addition property of equality is a fundamental technique in solving linear equations.

Algebraic Expressions

Understanding algebraic expressions is key to working through equations effectively. An algebraic expression involves numbers, variables, and arithmetic operations.

In our equation \(x + \frac{1}{3} = \frac{7}{3}\), we have expressions on each side. On the left side, \(x + \frac{1}{3}\) is an algebraic expression. The variable \(x\) is combined with a fraction, creating an expression that we process mathematically through addition or subtraction.

When solving, we treat the components of an expression distinctly. For example:

• Identify the operation and decide how to isolate the variable.
• Use properties of equality to manipulate the terms effectively.

Recognizing these expressions and their role in an equation is crucial. It allows us to break down and concentrate on the parts involving the variable, which in turn simplifies the equation to find the desired solution.

Checking Solutions

Once a solution is found, verifying it is not just a good practice but an essential step. Checking solutions ensures that no errors were made during calculations, and the solution truly satisfies the original equation.

For the exercise \(x+\frac{1}{3} = \frac{7}{3}\), we concluded with \(x = 2\). To check, we substitute \(x\) back into the original equation:

• Calculate \(2 + \frac{1}{3}\) which results in \(\frac{7}{3}\)
• Compare this with the original right side of the equation \(\frac{7}{3}\)

Both sides match, confirming our solution is correct.

Checking lets us confirm the correctness of our solution and is especially helpful in complex problems where mistakes are more likely. This step highlights that mathematics often involves a cycle of solving and verifying, reinforcing understanding and accuracy.