Problem 15 - Elementary and Intermediate Algebra

Question

Solve each equation and check the result. If an equation has no solution, so indicate. $$ \frac{2}{3}=\frac{1}{2}+\frac{x}{6} $$

Step-by-Step Solution

Answer

The solution is \( x = 1 \).

Step 1: Understand the Equation

We are given the equation \( \frac{2}{3} = \frac{1}{2} + \frac{x}{6} \). Our goal is to solve for \( x \). We will do this by isolating \( x \) on one side of the equation.

Step 2: Eliminate Fractions

To eliminate the fractions, find the least common denominator (LCD) of the fractions. The denominators are 3, 2, and 6. The LCD of these numbers is 6. Multiply every term in the equation by 6 to eliminate the denominators:\[ 6 \times \frac{2}{3} = 6 \times \frac{1}{2} + 6 \times \frac{x}{6} \]

Step 3: Simplify the Equation

Simplify each term after multiplying by 6:- \( 6 \times \frac{2}{3} = 4 \)- \( 6 \times \frac{1}{2} = 3 \)- \( 6 \times \frac{x}{6} = x \)The equation now becomes:\[ 4 = 3 + x \]

Step 4: Isolate the Variable

Subtract 3 from both sides of the equation to solve for \( x \):\[ 4 - 3 = 3 + x - 3 \]This simplifies to:\[ 1 = x \]

Step 5: Check the Solution

Substitute \( x = 1 \) back into the original equation to verify the solution:\[ \frac{2}{3} = \frac{1}{2} + \frac{1}{6} \]Find a common denominator for \( \frac{1}{2} \) and \( \frac{1}{6} \), which is 6, and verify:\[ \frac{1}{2} = \frac{3}{6} \] and \( \frac{1}{6} = \frac{1}{6} \), so \( \frac{3}{6} + \frac{1}{6} = \frac{4}{6} = \frac{2}{3} \).The left-hand side equals the right-hand side, so the solution is correct.

Key Concepts

Fractions

Fractions are parts of a whole, represented by two numbers separated by a line. The number at the top is the numerator, and the number at the bottom is the denominator. Fractions can be found in many algebraic equations, and they require understanding to manipulate correctly.
To solve equations involving fractions, you might need to add, subtract, multiply, or divide them. For example, in the expression \( \frac{2}{3} = \frac{1}{2} + \frac{x}{6} \), each fraction has a different denominator. That makes it trickier, but not impossible!
Fractions are useful in equations because they represent ratios and proportions, which means you can work with parts of a unit. Knowing how to handle fractions allows you to rearrange equations and find solutions efficiently. Remember, maintaining balance on both sides of an equation is key when dealing with these numbers. Once you master operations with fractions, solving for unknowns becomes much simpler.

Least Common Denominator

When dealing with fractions in equations, finding the least common denominator (LCD) is an essential step. The LCD helps you combine fractions by providing a common base for all terms involved.
For example, if you have fractions like \( \frac{1}{2} \) and \( \frac{x}{6} \), their denominators are 2 and 6. By determining their LCD, you can transform each fraction to have the same denominator, simplifying the process of solving equations.
To find the LCD, identify the least common multiple (LCM) of all the denominators. In our example, the denominators are 3, 2, and 6. The LCM of these numbers is 6, which becomes the LCD. Multiply each fraction by a factor that converts its denominator to the LCD.
Using the LCD, you eliminate fractions, which simplifies the equation and allows you to solve for the unknown variable with ease. This step ensures you don't have awkward fractional computations along the way, turning a potentially complex equation into something much more manageable.

Solving Equations

Solving equations, such as \( \frac{2}{3} = \frac{1}{2} + \frac{x}{6} \), involves a few important steps to find the value of the unknown variable, typically represented as \( x \). Solving an equation means determining the value(s) that make the equation true.
To solve equations that include fractions, start by eliminating the fractional components. This is often achieved by finding and using the least common denominator to create an equivalent equation without fractions. This makes the equation easier to handle.
Once fractions are eliminated, you can solve for the variable as you would in any other linear equation:

• Isolate the variable on one side of the equation by performing inverse operations. This may involve adding, subtracting, multiplying, or dividing both sides of the equation.
• Simplify the terms to reveal the solution.
• Finally, check your work. Substitute your solution back into the original equation to verify that it satisfies the equation.

In the provided equation, solving ultimately shows that \( x = 1 \). This process demonstrates how systematic handling of equations can reveal solutions, thoroughly preparing you for solving more complex algebraic problems in the future.