Problem 76 - Introductory Algebra for College Students

Question

Solve each equation or system of equations. $$\frac{7}{x+2}+\frac{2}{x+3}=\frac{1}{x^{2}+5 x+6}$$

Step-by-Step Solution

Answer

The solution to the equation is \( x = -\frac{8}{3} \)

Step 1: Identifying the common denominator

First, notice all the denominators \(x + 2\), \(x + 3\), and \(x^2 + 5x + 6\). The third denominator \(x^2 + 5x + 6\) can be factored as \((x + 2) * (x+ 3)\). Thus, the common denominator is \((x + 2) * (x+ 3)\).

Step 2: Eliminate the Fractions

Next, multiply each term in the equation by the common denominator \((x + 2) * (x + 3)\) to eliminate the fractions: \[ (x + 2)*(x + 3)*\frac{7}{x+2} + (x + 2)*(x + 3)*\frac{2}{x+3} = (x + 2)*(x +3)*\frac{1}{x^2+5x+6} \] Simplify this to: \[ 7*(x + 3) + 2*(x + 2) = 1\]

Step 3: Simplify and Solve

Then simplify the equation: \[ 7x + 21 + 2x + 4 = 1 \] Combine like terms to find the value of x: \[ 9x + 25 = 1 \] Subtract 25 from both sides: \[ 9x = -24 \] Finally, divide by 9: \[ x = - \frac{24}{9} \]

Step 4: Reduce to Lowest Terms

Reduce the fraction to its lowest terms to get the final solution: \[x = -\frac{8}{3}\]

Key Concepts

Common Denominator

When faced with an equation involving multiple fractions, identifying a common denominator is a crucial first step. This common denominator serves as a shared base that allows you to combine the fractions or eliminate them entirely, which simplifies the equation-solving process.

In the example \( \frac{7}{x+2} + \frac{2}{x+3} = \frac{1}{x^2 + 5x + 6} \), we observe that the first two denominators are \(x + 2\) and \(x + 3\), while the third denominator can be factored into \( (x + 2) * (x + 3) \), which reveals it is the product of the first two. Therefore, the common denominator for all terms is indeed \( (x + 2) * (x + 3) \). By using this common denominator, we can then eliminate the fractions, making our equation easier to handle.

Equation Simplification

Once the common denominator is found, equation simplification involves manipulating the equation into a simpler form that is easier to solve. This usually starts by multiplying every term by the common denominator to get rid of the fractions, as in this case:

\[\begin{align*} (x + 2)&(x + 3)\left(\frac{7}{x+2}\right) + (x + 2)(x + 3)\left(\frac{2}{x+3}\right) &= (x + 2)(x +3)\left(\frac{1}{x^2+5x+6}\right) \end{align*}\]
This leads to a much simpler linear equation that usually only requires combining like terms and basic arithmetic to solve. Simplification is essential as it converts complex, fractional equations into straightforward algebraic expressions.

Algebraic Fractions

Algebraic fractions are simply fractions that contain variables. They can initially seem daunting due to the presence of unknowns in the numerators, denominators, or both. However, like regular fractions, you can work with them effectively once you understand the concepts of common denominator and simplification.

In the provided exercise, you encounter algebraic fractions such as \( \frac{7}{x+2} \) and \( \frac{2}{x+3} \). As in standard fraction addition, you would typically aim to have the same denominator before combining them. However, in solving equations, we opt to eliminate these fractions early on through multiplication by the common denominator, which greatly simplifies the process of finding the solution.

Fraction Reduction

Fraction reduction is a process of simplifying a fraction to its lowest terms by cancelling out common factors from the numerator and the denominator. This is an important final step in solving equations as it ensures the solution is presented in its simplest form.

In our example, the simplified form of \( x \) was found to be \( -\frac{24}{9} \). However, both 24 and 9 share a common factor of 3. Dividing the numerator and the denominator by this common factor reduces the fraction to \( -\frac{8}{3} \), which is the final answer in its lowest terms. Knowing how to reduce fractions not only confirms the correctness of a solution but also demonstrates a comprehensive understanding of number properties.