Problem 83 - College Algebra with Modeling and Visualization

Question

Solve the equation. $$ \frac{x+1}{x}=2 $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 1\).

Step 1: Eliminate the Fraction

Start by multiplying both sides of the equation \(\frac{x+1}{x} = 2\) by \(x\) to get rid of the fraction. This gives us:\[x+1=2x\]

Step 2: Simplify the Equation

Subtract \(x\) from both sides to isolate the variable on one side:\[1 = 2x - x\]Simplifying gives:\[1 = x\]

Step 3: Verify the Solution

Substitute \(x = 1\) back into the original equation to ensure it satisfies the equation:\[\frac{1+1}{1} = 2\]Simplifying the left hand side gives:\[\frac{2}{1} = 2\]Since both sides of the equation are equal, \(x = 1\) is confirmed as the correct solution.

Key Concepts

Solving Linear Equations

Linear equations are a fundamental part of algebra involving variables raised to the power of one. These equations can often be solved using straightforward steps. In the problem \( \frac{x+1}{x} = 2 \), we work on the equation involving a variable, \( x \), that needs to be solved.
To solve a linear equation, our primary goal is to isolate the variable on one side of the equation. This can be achieved through:

• Adding or subtracting terms on both sides.
• Multiplying or dividing by a constant.
• Using operations that maintain equality, like distributing or factoring.

In our example, after eliminating the fraction, we simplified the equation and subtracted \( x \) from both sides. This led to the simplified equation \( 1 = x \). The objective is always to find out what value(s) of \( x \) satisfy the equation when plugged back into it. This approach ensures that the solution identifies the specific value for the unknown variable.

Elimination of Fractions

Fractions can complicate solving equations, so it's often helpful to eliminate them. The fraction in \( \frac{x+1}{x} = 2 \) requires a technique to simplify the equation, making it more manageable to solve.
To eliminate fractions and make equations easier to manipulate:

• Multiply every term by a common denominator. This helps convert fractions into whole numbers.
• Ensure each fraction term has its denominator canceled out properly.

In our exercise, multiplying both sides by \( x \) eradicated the fraction, effectively transforming the equation to \( x+1 = 2x \). This new form makes isolating \( x \) simpler, enabling a direct approach to solving linear equations.

Equation Verification

Verification is the final step that ensures the solution is correct. Once we find the potential solution \( x = 1 \), it's crucial to check that it satisfies the initial equation.
To verify a solution:

• Substitute the value back into the original equation.
• Simplify both sides of the equation.
• Check if both sides are equal, confirming the solution fits the equation.

For our problem, substituting \( x = 1 \) gives us \( \frac{1+1}{1} = 2 \). Simplifying, both sides equal \( 2 \), securing our solution. Always verify to avoid errors and ensure absolute accuracy in solving equations. This step verifies that the computations leading to the result are correct and applicable to the original equation.