Problem 51 - Precalculus

Question

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

Step-by-Step Solution

Answer

The solution for \(x\) is \(x = 1\) if \(a \neq b\).

Step 1: Identify Like Terms

Look at the equation \(a x + b = b x + a\). We need to identify like terms that we can simplify or move to one side of the equation.

Step 2: Move Terms Involving x to One Side

Subtract \(b x\) from both sides to get rid of the variable \(x\) on one side: \(a x + b - b x = a\). The equation simplifies to \(a x - b x + b = a\).

Step 3: Factor Out x

Now, factor \(x\) from the terms on the left: \((a - b)x + b = a\).

Step 4: Isolate x

Subtract \(b\) from both sides to isolate the term with \(x\): \((a - b)x = a-b\).

Step 5: Solve for x

Divide both sides by \((a - b)\) to solve for \(x\):\[x = \frac{a - b}{a - b}\].

Step 6: Simplify Expression

Since \(\frac{a-b}{a-b} = 1\) for \(a eq b\), the final expression for \(x\) is \(x = 1\).

Key Concepts

Like Terms

When solving algebraic equations, one of the first things you want to do is identify and work with like terms. Like terms are terms that have the same variable raised to the same power. In our example equation, \(a x + b = b x + a\), you should first notice that the terms \(a x\) and \(b x\) both contain the variable \(x\). These are like terms because they share the common variable.Identifying like terms helps you to simplify the algebraic expression effectively. When terms are similar, they can be combined or moved around in the equation. Using properties of equality, you can manipulate these terms by adding, subtracting, or factoring them out to make the equation easier to solve. For example, if you have \(a x + b x\), you can combine these to \((a + b)x\).Earlier recognition of like terms saves time and reduces mistakes. It’s a fundamental skill that makes the entire problem-solving process smoother and more intuitive.

Variable Isolation

To solve for a specific variable in an equation, an essential step is isolating that variable on one side of the equation. This process is known as variable isolation. In our given equation \(a x + b = b x + a\), the aim is to solve for \(x\) which requires all terms involving \(x\) to be brought to one side of the equation.

• Subtract \(b x\) from both sides: This simplifies the equation by removing \(x\) from the right side, resulting in \(a x - b x + b = a\).
• Factor out the \(x\): From the expression \(a x - b x\), factor out \(x\) to get \((a - b)x + b = a\).

These steps help in isolating \(x\) effectively. Remember, the goal is to have \(x\) on one side so you can easily determine its value. Once \(x\) is isolated, solving the rest of the equation becomes straightforward.

Equation Simplification

After isolating the variable, the final piece of solving an equation is simplifying it to its most basic form. This ensures that you have solved for the exact value of the variable without unnecessary complexity. Consider the worked example:The equation \((a-b)x + b = a\) focuses on simplifying by removing the +\(b\) term:
1. Subtract \(b\) from both sides: \((a - b)x = a - b\).
2. Divide both sides by \(a - b\) to solve for \(x\):\[x = \frac{a - b}{a - b}\\]This process uses basic algebraic laws of operations to reach the simplest form: \(x = 1\). As long as \(a\) is not equal to \(b\), this calculation holds true. Always double-check your simplification, as simplification errors can lead to incorrect solutions.Simplifying often also means checking for conditions where division by zero could occur, ensuring the solution is valid. By adhering to mathematical operations correctly, the process of simplification confirms the integrity of your solution.