Problem 29 - College Algebra

Question

Exercises \(17-30\) contain equations with constants in denominators. Solve each equation. $$ \frac{x+1}{3}=5-\frac{x+2}{7} $$

Step-by-Step Solution

Answer

\(x = 10.4\)

Step 1: Eliminate fractions

Start by multiplying each individual term by the common denominator (which is 21 here) to eliminate fractions. Hence, the equation becomes: \(7(x+1) = 21*5 - 3*(x+2)\)

Step 2: Simplify

Simplify both sides of the equation to get: \(7x + 7 = 105 - 3x - 6\)

Step 3: Rearrange

Rearrange the equation by getting terms involving \(x\) to one side and constants to the other to get: \(7x + 3x = 105 - 7 + 6\)

Step 4: Simplify again

Simplify both sides once again to get: \(10x = 104\).

Step 5: Solve for x

Finally, isolate \(x\) by dividing both sides by 10: \(x = 104/10 = 10.4\)

Key Concepts

Fractions

Fractions are mathematical expressions representing the division of one quantity by another. In our example, the equation involves fractions with the terms \(\frac{x+1}{3}\) and \(-\frac{x+2}{7}\). The fraction bar acts as a division symbol that separates the numerator and the denominator. Here’s how to interpret each part:

• The numerator is the top part of the fraction.
• The denominator is the bottom part, showing into how many parts the whole is divided.

Fractions can be tricky to work with, especially in equations. They often need special attention like finding a common denominator or converting them to a simpler form for easier manipulation.

Common Denominator

Finding a common denominator is a crucial step when working with fractions in equations. It allows you to eliminate fractions and simplify the equation. The common denominator is a multiple of all the denominators in the fractions you are dealing with. In the equation \(\frac{x+1}{3}=5-\frac{x+2}{7}\), the denominators are 3 and 7.

• The smallest common multiple of these numbers is 21.

By multiplying every term by this common denominator, we get rid of the fractions and transform the equation into one without fractions. In this case, it turns into \(7(x+1) = 21*5 - 3*(x+2)\). The transformation from fractional terms to straightforward algebra simplifies the process of solving the equation, avoid fractional complexities, thus bringing clarity to calculations.

Algebraic Rearrangement

Algebraic rearrangement is the process of manipulating equations to simplify them or move terms to where they belong. This involves changing the order or grouping of terms without changing the equality. In our equation, after getting rid of the fractions, we simplify both sides and move terms to create order.

• First, simplify the expressions \(7x + 7\) and \(105 - 3x - 6\).
• Then, rearrange by gathering all terms with \(x\) on one side and the constants on the other.
• The equation thus balances as \(7x + 3x = 105 - 7 + 6\).

This rearrangement is crucial for the next step, which is isolating the variable. It reduces equations to a form that is simpler to solve with clear pathways to find our goal variable.

Isolating Variables

Isolating the variable is a key step in solving equations. It means getting the variable (usually \(x\)) by itself on one side of the equation. This process allows us to solve for the unknown value easily. Here’s how we can do it:

• Start with a balanced equation, like \(10x = 104\).
• We aim to have \(x\) alone to find its value, so we divide both sides by 10.
• Doing so gives \(x = \frac{104}{10} = 10.4\).

Isolating variables involves reversing any operations that you performed on \(x\). Here, division was used to nullify the multiplication, thus isolating \(x\). This approach is crucial in algebra as it simplifies solving even the most complex equations step by step until the unknown variable is found.