Problem 34 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the equation (if possible). $$\frac{x-11}{x}=\frac{x-9}{x}+2$$

Step-by-Step Solution

Answer

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

Step 1: Simplify the equation

Combine the fractions into one on either side of the equation. We observe that the denominators on the left side and the right side are the same. So the equation simplifies to \(x - 11 = x - 9 + 2x\).

Step 2: Group similar terms

Group similar terms together. Upon combining the similar terms, the equation becomes \(-11 + 9 = 2x\).

Step 3: Simplify to get the value of x

By simplifying the left side, we obtain that \(x = -2/2\). So, \(x = -1\).

Key Concepts

Simplifying Algebraic Expressions

Simplifying algebraic expressions is a fundamental step in solving equations. At its core, the idea is to make the expression as straightforward as possible. In this exercise, we started with fractions on both sides of the equation:

• On the left, \( \frac{x-11}{x} \).
• On the right, \( \frac{x-9}{x} + 2 \).

Notice that both fractions have the common denominator \(x\). This allows us to simplify by recognizing terms that can be eliminated or combined. By simplifying, \(2\) can be viewed as \(\frac{2x}{x}\) so it aligns perfectly with the common denominator. After substitution, the right side becomes \( \frac{x-9+2x}{x} \), which simplifies to \( x - 11 = x - 9 + 2x \) once fractions are eliminated by multiplication by the common denominator. Removing fractions can simplify equations considerably, but it's essential to ensure all steps maintain the equation's balance.

Combining Like Terms

Combining like terms is another crucial step in simplifying an equation. Like terms are terms that involve the same variable raised to the same power. In our equation, like terms are those that have the variable \(x\).In the simplified equation, \( x - 11 = x - 9 + 2x \), you can see that terms with \(x\) are present on both sides:

• On the left, there's \( x \).
• On the right, there are two terms \( x \) and \( 2x \).

We can combine these like terms (\( x \) and \( 2x \)) to make equation easier to solve. The right side becomes \(3x - 9\). Next, isolate \(x\) by performing operations on both sides, ultimately leading to the solution. When solved correctly for terms \(-11 + 9 = 2x\), now simplifies further to the negative form of the result \(-2 = 2x \), ensuring the clear presentation of outcome.

Fractions in Equations

Working with fractions in equations can be intimidating, but understanding them makes the process smoother. In our problem, both sides contain fractions with the same denominator. This allows for easier manipulation. Having common denominators means the variable can be ignored while combining the numerators. The key to simplifying fractions in equations is to eliminate the denominators by multiplying all parts of the equation by the denominator itself.In this case, consider the fraction: \( \frac{x-11}{x} = \frac{(x-9) + 2x}{x} \).

• Observing that we can multiply through by \(x\) immediately balances out the equation by making the denominator disappear.
• This leaves us easy access to solve the equation through traditional arithmetic means.

This method of clearing fractions simplifies what can be a frustrating process, allowing us to focus on integer values to reach the solution faster, without errors typically made when dealing with numerators and denominators separately.