Solve each equation. 10/(4x-4) = 1 /(1-x)

Verified step by step guidance

1. Start by rewriting the equation clearly: \(\frac{10}{4x - 4} = \frac{1}{1 - x}\).
2. Identify the denominators on both sides: \$4x - 4\( and \)1 - x\(. Notice that \)4x - 4\( can be factored as \)4(x - 1)$.
3. To eliminate the fractions, multiply both sides of the equation by the least common denominator (LCD), which is \$4(x - 1)(1 - x)\(. Remember that \)x - 1\( and \)1 - x$ are negatives of each other.
4. After multiplying both sides by the LCD, simplify the resulting equation by canceling denominators and carefully handling the signs.
5. Solve the resulting linear equation for \(x\), then check your solution(s) to ensure they do not make any denominator zero in the original equation.

Key Concepts

Solving Linear Equations

Solving linear equations involves finding the value of the variable that makes the equation true. This typically requires isolating the variable on one side by performing inverse operations such as addition, subtraction, multiplication, or division.

Working with Fractions in Equations

When equations contain fractions, it is important to find a common denominator or use cross-multiplication to eliminate the fractions. This simplifies the equation and makes it easier to solve for the variable.

Distributive Property and Simplification

The distributive property allows you to multiply a single term across terms inside parentheses. Simplifying expressions by distributing and combining like terms is essential before isolating the variable in an equation.