Problem 77 - Algebra and Trigonometry

Question

Combine the types of equations we have discussed in this section. Solve equation. Then state whether the equation is an identity, a conditional equation, or an inconsistent equation. $$\frac{4}{x-2}+\frac{3}{x+5}=\frac{7}{(x+5)(x-2)}$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = -1\). The equation is a conditional equation because it is only valid for \(x = -1\).

Step 1: Identify Common Denominator

First find the common denominator for all fractions in the equation. In this case, the common denominator is \((x-2)(x+5)\).

Step 2: Multiply Each Term by the Common Denominator

Next, multiply each term in the equation by the common denominator and simplify. When multiplying by the common denominator, the fractions are 'cleared', resulting in the equation \(4(x+5) + 3(x-2) = 7\).

Step 3: Simplify the Equation

Solve the simplified equation. This involves expanding the brackets, adding like terms and then isolating the variable. So, expanding the brackets gives \(4x+20+3x-6=7\), which simplifies to \(7x+14=7\). Subtracting 14 from both sides of the equation gives \(7x= -7\), and dividing both sides by 7 results in \(x=-1\).

Step 4: Check the Solution

Having found a solution, the next step is to substitute it back into the original equation and check if it satisfies the equation. Plugging \(x=-1\) into \(\frac{4}{x-2}+\frac{3}{x+5}-\frac{7}{(x+5)(x-2)}=0\), we find that it results in \(0=0\). Therefore, \(x=-1\) is indeed a solution, making the equation conditional.

Key Concepts

Understanding the Common Denominator

When faced with rational equations like \(\frac{4}{x-2}+\frac{3}{x+5}=\frac{7}{(x+5)(x-2)}\), the first step is often to find the common denominator. This is crucial because it allows you to eliminate the fractions, simplifying the process of solving the equation.
To determine a common denominator, look for the product of the different denominators in the equation. In this exercise:

• \((x-2)\) and \((x+5)\) are the distinct denominators.
• The common denominator is found by multiplying these: \((x-2)(x+5)\).

Choosing the right common denominator lets us "clear" the fractions by multiplying each term in the equation by this value. This step transitions a rational equation into a more straightforward linear form, making subsequent steps smoother.

Simplifying Equations After Clearing Fractions

Once the common denominator has been used to clear the fractions, you will be left with an equation to simplify. In this exercise, multiplying through transforms the equation into: \[ 4(x+5) + 3(x-2) = 7 \].
Next, you need to expand and simplify this equation:

• Distribute the numbers outside the brackets: \(4(x+5)\) becomes \(4x + 20\) and \(3(x-2)\) becomes \(3x - 6\).
• Combine like terms to simplify: \(4x + 3x + 20 - 6 = 7\), which results in \(7x + 14 = 7\).

The goal here is to isolate the variable \(x\) by systematically combining terms and performing arithmetic operations to simplify the equation to something easily solvable.

Solving Rational Equations

Now that we've simplified the equation, the process of solving it can begin. From our simplified form \(7x + 14 = 7\), our aim is to isolate \(x\).
The steps are simple:

• Subtract 14 from both sides to get \(7x = -7\).
• Divide both sides by 7 to isolate \(x\), resulting in \(x = -1\).

After finding a solution, it's important to check it by substituting back into the original equation to see if it holds true. Substituting \(x = -1\) confirms the solution, as it balances the equation.
This type of solution indicates a conditional equation, meaning it is true for specific values of \(x\). Understanding these steps ensures confidence when tackling similar rational equations.