Problem 26 - College Algebra

Question

Exercises \(17-30\) contain equations with constants in denominators. Solve each equation. $$ \frac{x+1}{4}=\frac{1}{6}+\frac{2-x}{3} $$

Step-by-Step Solution

Answer

The solution for the equation is \(x=1\).

Step 1: Clear the Fractions

Multiply each term of the equation by 12 (the least common multiple of the denominators) to clear out the fractions. This operation will result in: \(12*\frac{x+1}{4}=12*\frac{1}{6}+12*\frac{2-x}{3}\) which simplifies to \(3x+3=2+4*(2-x)\).

Step 2: Simplify the Equation

Simplify the equation by carrying out the multiplication: \(3x+3=2+8-4x\). This simplifies further to \(3x+3=10-4x\).

Step 3: Collect Like Terms

Bring the terms involving 'x' to one side of the equation and constants to the other. This results in: \(3x+4x=10-3\), simplifying to \(7x=7\).

Step 4: Solve for x

Finally, solve for ‘x’ by dividing both sides of the equation by 7 which gives \(x=1\).

Key Concepts

Clearing Fractions in Equations

Solving equations that contain fractions can be simplified by 'clearing' the fractions. Clearing fractions means to eliminate them from the equation, which makes the equation easier to work with. This is done by finding a common multiple of all the denominators in the equation and multiplying each term by that number.

This process transforms the fractional coefficients into whole numbers. For example, consider the equation \(\frac{x+1}{4} =\frac{1}{6} + \frac{2-x}{3}\). To clear the fractions, you would multiply each term by the least common multiple (LCM) of the denominators, which in this case is 12. This results in \(12*\frac{x+1}{4}=12*\frac{1}{6}+12*\frac{2-x}{3}\), which simplifies to a new equation without fractions: \(3x+3=2+4*(2-x)\).

It's crucial to multiply every term by the LCM to maintain the balance of the equation. Once the fractions are cleared, the equation becomes more straightforward to solve by applying additional algebraic principles.

Least Common Multiple

The least common multiple (LCM) is a key concept when clearing fractions in an equation. It refers to the smallest number that is a multiple of two or more denominators. The LCM allows you to find a common ground where all fractions can be 'cleared', or converted to whole numbers, without changing the value of the equation.

To find the LCM of the denominators, first list out the multiples of each denominator until you find the smallest multiple that appears in all lists. For example, if the denominators are 4, 6, and 3—the multiples will be:

• 4: 4, 8, 12, 16, 20, ...
• 6: 6, 12, 18, 24, ...
• 3: 3, 6, 9, 12, 15, ...

The smallest common multiple in all three lists is 12. Therefore, the LCM of 4, 6, and 3 is 12. Using the LCM to clear the fractions ensures that all the original ratios are preserved while simplifying the solving process.

Collecting Like Terms

Once an equation is free of fractions, the next step is to group or 'collect' like terms. This means moving all terms containing the variable (such as 'x') to one side of the equation and all the constant terms to the other side. This step helps to condense the equation and get the variable on its own.

For instance, after clearing fractions and simplifying the given example equation, you get \(3x+3=10-4x\). Collecting like terms involves two steps: adding \(4x\) to both sides to move all the 'x' terms to the left, and subtracting 3 from both sides to move the constant to the right. This yields \(3x+4x=10-3\), which further simplifies to \(7x=7\). When like terms are together, it is easier to see the relationship between the variable and constants, paving the way to finding the solution.