Problem 19 - Algebra for College Students

Question

Find the solution set to each equation. $$x+1+\frac{2 x-5}{x-5}=\frac{x}{x-5}$$

Step-by-Step Solution

Answer

The solution is \( x = -2 \).

Step 1: Move the rational terms to one side

Start by isolating the rational terms on one side of the equation. Subtract \(\frac{x}{x-5}\) from both sides of the equation: \[ x + 1 + \frac{2x - 5}{x-5} - \frac{x}{x-5} = 0 \]

Step 2: Combine the rational terms

Combine the terms involving fractions: \[ \frac{2x - 5}{x-5} - \frac{x}{x-5} = \frac{2x - 5 - x}{x-5} = \frac{x - 5}{x-5} = 1 \] So the equation now becomes: \[ x + 1 + 1 = 0 \]

Step 3: Simplify

Now simplify the equation: \[ x + 2 = 0 \]

Step 4: Solve for \(x\)

Subtract 2 from both sides to solve for \( x \): \[ x = -2 \]

Key Concepts

Isolating Variables

When solving rational equations, the first step often involves isolating variables. This means you want to get the variable you're solving for on one side of the equation. In our example, we start by moving the rational terms to one side:
$$x + 1 + \frac{2x - 5}{x-5} = \frac{x}{x-5}$$
To isolate the variables, we subtract \frac{x}{x-5}\ from both sides:
$$x + 1 + \frac{2x - 5}{x-5} - \frac{x}{x-5} = 0$$
. Isolating variables simplifies solving the equation because it groups similar terms together, making the next steps clearer.

Combining Fractions

In rational equations, combining fractions is essential when you find you're dealing with terms having the same denominator. In this case, after isolating the variable, you have:
$$\frac{2x - 5}{x-5} - \frac{x}{x-5}$$
Since both fractions share the same denominator, you can easily combine them into one:
$$\frac{2x - 5 - x}{x-5} = \frac{x - 5}{x-5}$$.
. Always remember to combine fractions by dealing with their numerators and keeping the common denominator. This step simplifies the equation significantly, preparing it for solving.

Simplifying Equations

Simplifying equations is crucial in problem-solving. It involves reducing the equation to its simplest form. From the previous step, we had:
$$\frac{x - 5}{x-5}$$
Since the numerator and the denominator are the same, this fraction simplifies to 1:
$$\frac{x - 5}{x-5} = 1$$.
. This turns our equation into a much simpler form:
$$x + 1 + 1 = 0$$
, which simplifies further to:
$$x + 2 = 0$$.
Simplifying equations can be thought of as cleaning up your workspace. It makes the final solving steps much more straightforward.

Solving Linear Equations

The final step in many problems like this is solving a linear equation. Once we've simplified, we get to a point where:
$$x + 2 = 0$$
Now, you just have to solve for the variable by isolating it:
Subtract 2 from both sides:
$$x = -2$$
. Solving linear equations usually involves basic operations: addition, subtraction, multiplication, or division. It's about isolating the variable to find its value. And there you go! The solution is neatly found after all these steps:
$$x = -2$