Question

If \(\frac{\mathrm{x}}{(\mathrm{x}+2)(\mathrm{x}+3)}=\frac{\mathrm{A}}{\mathrm{x}+3}+\frac{\mathrm{B}}{\mathrm{x}+2}\), then \(\mathrm{A}-\mathrm{B}\) is \(=\) ____. (1) 4 (2) 5 (3) \(-6\) (4) 2

Short answer

Question: Calculate the value of A - B in the partial fraction decomposition of the given equation: \(\frac{x}{(x+2)(x+3)} = \frac{A}{x+2} + \frac{B}{x+3}\). Answer: 5

Step-by-step solution

Clear the denominators

Multiply both sides of the equation by \((x+2)(x+3)\): \(x = A(x+2) + B(x+3)\)

Solve for A and B

Set the numerators equal to one another: \(x = Ax + 2A + Bx + 3B\) Now, we can isolate \((A+B)x\) and the constant terms \((2A+3B)\): \(x = (A + B)x + (2A + 3B)\) Since this equation must be true for all values of \(x\), we can equate the coefficients of the powers of \(x\): \(A+B = 1\) and \(2A + 3B=0\)

Solve the system of equations

We have a system of two linear equations: 1. \(A+B =1\) 2. \(2A + 3B = 0\) We can eliminate \(A\) by multiplying the first equation by 2 and subtracting it from the second equation: \(3B - 2B = -2 \Rightarrow B = -2\) Now we substitute the value of \(B\) into the first equation: \(A - 2 = 1 \Rightarrow A = 3\)

Calculate A - B

Now that we have found \(A=3\) and \(B=-2\), we can calculate \(A-B\): \(A - B = 3 - (-2) = 3 + 2 = 5\) The correct answer is \((2)5\).

Key concepts

Linear Equations

Linear equations are algebraic expressions of the form \(ax + b = 0\), where \(a\) and \(b\) are constants. These equations are called 'linear' because they form a straight line when graphed on a coordinate plane. In the context of solving partial fraction decomposition, linear equations help us express complicated expressions as a sum of simpler fractions.
This involves finding values for unknown coefficients by setting up linear equations. The beauty of linear equations is their simplicity; solving them involves straightforward algebraic manipulation.

• Identify similar terms and group them.
• Use algebraic operations to isolate variables.

Mastering linear equations is a fundamental step in tackling more complex algebraic problems.

Equating Coefficients

Equating coefficients is a method used to solve equations by making the coefficients of like terms on both sides equal. When dealing with partial fraction decomposition, equating coefficients allows us to find unknown constants by comparing corresponding coefficients of polynomial expressions.
This technique is crucial because it transforms an equation into a simpler set of equations that can be solved independently. Here's how it typically works:

• First, expand any expressions to reveal all terms clearly.
• Next, compare the coefficients of each power of \(x\) on both sides.

Using equating coefficients simplifies solving even complicated algebraic equations, as it breaks down complex expressions into manageable parts.

System of Equations

A system of equations is a collection of two or more equations with the same set of variables. In the exercise, we turned an equation into a system by equating the coefficients of similar terms. This is a crucial strategy in algebra for solving multiple unknowns simultaneously.
Solving a system might require techniques such as substitution, elimination, or using matrices. Here's a quick look at the steps:

• Identify the equations that form the system.
• Use substitution or elimination to simplify the system into single-variable equations.

A system of equations gives deep insights into how different equations relate and interact, allowing us to solve for multiple unknowns at once.

Simplifying Algebraic Fractions

Simplifying algebraic fractions involves reducing them to a simpler or more manageable form. When dealing with partial fractions, simplifying is necessary to break a complex fraction into a sum of simpler ones. This process is essential to understand, as it makes further calculations easier and clearer.
Here's how to simplify algebraic fractions:

• Start by factoring both the numerator and the denominator.
• Cancel out common factors.
• Recombine fractions if necessary to achieve the simplest form.

Simplifying helps in easily integrating or differentiating fractions, and is a key skill for solving complex equations.