Problem 50 - Precalculus

Question

Solve each equation for \(x\) in terms of the other letters. $$a x+b=b x-a$$

Step-by-Step Solution

Answer

\(x = \frac{-a - b}{a-b} \)

Step 1: Isolate Terms with 'x' on One Side

Start by getting all terms with the variable \(x\) on one side of the equation. Move \(b x\) to the left side by subtracting \(b x\) from both sides.\[ a x - b x + b = -a \]

Step 2: Combine Like Terms

Combine the terms with \(x\) on the left side of the equation.\[ (a-b)x + b = -a \]

Step 3: Isolate the \(x\) Variable

Subtract \(b\) from both sides to further isolate \(x\).\[ (a-b)x = -a - b \]

Step 4: Solve for \(x\)

To completely isolate \(x\), divide both sides of the equation by \((a-b)\), assuming \(a eq b\).\[ x = \frac{-a - b}{a-b} \]

Key Concepts

Solving Equations

When solving equations, the main goal is to find the value of the variable that makes the equation true. This process involves several steps, including moving terms from one side of the equation to the other and transforming the equation until the variable is isolated.
For example, in the equation given, move terms around so like terms are grouped together, making it easier to solve for the variable.

• Start by deciding which variable you want to isolate (usually all terms with the variable on one side of the equation).
• Then, perform arithmetic operations such as addition, subtraction, multiplication, or division symmetrically to maintain equality.
• Ensure you follow the correct order of operations to avoid mistakes.

Once the equation is simplified and rearranged, you can find the variable's value that balances the equation.

Variable Isolation

Variable isolation is essential for finding the solution of an equation. It refers to getting the variable you are solving for alone on one side of the equation. This is especially important in equations where more than one term contains the variable of interest.
In the example provided, achieving variable isolation required moving all terms involving 'x' to one side and the constants to the other. This was done by:

• Subtracting terms to cancel out constants or other variable terms from the side where you are isolating the main variable.
• Simplifying complex expressions by performing arithmetic operations.

Thus, when isolating 'x,' ensure the end step results in only 'x' equals something, as neat and simple as possible.

Combining Like Terms

Combining like terms refers to simplifying an expression by adding or subtracting coefficients that have the same variable component.
In the equation originally given, there are two terms involving 'x,' specifically 'ax' and 'bx' on different sides. To combine these terms effectively:

• Move them to the same side of the equation.
• Add or subtract the coefficients (numbers in front of the variable) where applicable.

This makes the equation a lot easier to solve, as it reduces the number of separate terms. In this exercise, by combining like terms, we simplified _ ax - bx_ to _(a-b)x_ to help isolate the variable efficiently.