Solve each equation for x. x/(a-1) = ax+3

Verified step by step guidance

1. Start with the given equation: \(\frac{x}{a} - 1 = ax + 3\).
2. To eliminate the fraction, multiply every term on both sides of the equation by \(a\) to get rid of the denominator: \(a \times \left( \frac{x}{a} - 1 \right) = a \times (ax + 3)\).
3. Simplify both sides: on the left, \(a \times \frac{x}{a} = x\) and \(a \times (-1) = -a\); on the right, \(a \times ax = a^2 x\) and \(a \times 3 = 3a\). So the equation becomes \(x - a = a^2 x + 3a\).
4. Next, collect all terms involving \(x\) on one side and constants on the other side. For example, subtract \(a^2 x\) from both sides and add \(a\) to both sides: \(x - a^2 x = 3a + a\).
5. Factor out \(x\) on the left side: \(x(1 - a^2) = 4a\). Finally, solve for \(x\) by dividing both sides by \((1 - a^2)\): \(x = \frac{4a}{1 - a^2}\).

Key Concepts

Solving Linear Equations

Solving linear equations involves isolating the variable on one side of the equation using inverse operations such as addition, subtraction, multiplication, and division. The goal is to find the value of the variable that makes the equation true.

Working with Fractions in Equations

When an equation contains fractions, it is often helpful to eliminate the denominators by multiplying both sides by the least common denominator. This simplifies the equation and makes it easier to solve for the variable.

Distributive Property

The distributive property allows you to multiply a single term by each term inside parentheses, expressed as a(b + c) = ab + ac. This property is useful for expanding expressions and simplifying equations before solving.