Problem 85 - College Algebra with Modeling and Visualization

Question

Solve the equation. $$ \frac{1-x}{3 x-1}=-\frac{3}{5} $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = -\frac{1}{2}\).

Step 1: Cross-multiply

To eliminate the fractions, perform cross-multiplication. Multiply the numerator of the left fraction by the denominator of the right fraction and vice versa: \((1 - x) \cdot 5 = (3x - 1) \cdot (-3)\).

Step 2: Distribute the terms

Distribute the terms in both products from the previous step: \(5 - 5x = -9x + 3\).

Step 3: Simplify the equation

Move all the terms involving \(x\) to one side and constant terms to the opposite side: \(-5x + 9x = 3 - 5\).

Step 4: Combine like terms

Combine the terms with \(x\) and the constant terms: \(4x = -2\).

Step 5: Solve for x

Divide both sides of the equation by 4 to solve for \(x\): \(x = \frac{-2}{4}\). Simplify the fraction: \(x = -\frac{1}{2}\).

Key Concepts

Equations

An equation is a mathematical statement that shows the equality between two expressions. Equations can include numbers, variables, and mathematical operations. They often have a variable that you need to solve for, like the equation in our problem. Here, the goal is to find the value of \(x\) that makes the equation true. Think of equations as a balance scale: whatever you do to one side, you must do to the other. This balance is crucial for solving the equation correctly. Variables like \(x\) represent unknown values that we find using various methods, such as cross-multiplication or simplification. Understanding equations is fundamental in algebra, as it forms the basis for most algebraic problem-solving.

Cross-multiplication

Cross-multiplication is a method used to solve equations with fractions. When you have a proportion or an equation involving two fractions set equal to each other, cross-multiplication can clear the fractions and simplify the equation. This process involves multiplying the numerator of one fraction by the denominator of the other.In our problem: - The equation \(\frac{1-x}{3x-1} = -\frac{3}{5}\) was solved by cross-multiplying. - Multiply \((1-x)\) by 5 and \((3x-1)\) by \(-3\).The main advantage of this technique is that it eliminates the fractions, making the equation easier to work with. This reduction simplifies subsequent algebraic operations, such as distributing and combining like terms.

Simplifying equations

Simplifying equations is about reducing complexity to easily solve them. After cross-multiplying, the next step is often distributing terms and combining like terms. Let's look at the equation after cross-multiplying: - We have \(5 - 5x = -9x + 3\) from the operation.Next steps include:- Distributing terms across parentheses.- Grouping all \(x\) terms on one side and constants on the other. For example:- Moving terms: Adjust \(-5x + 9x = 3 - 5\).- Combine like terms to get \(4x = -2\).This step-by-step reduction continues until the equation is simplified enough to solve for \(x\) effectively. The goal is to have a clear, straightforward expression where the solution becomes apparent, like \(x = -\frac{1}{2}\) after dividing both sides by 4.