Problem 112 - Introductory Chemistry Atoms First

Question

Why can't you multiply just one side of an equation by something when algebraically rearranging the equation?

Step-by-Step Solution

Answer

When algebraically rearranging an equation, it is crucial to maintain the balance of the equation. If you multiply just one side of the equation, it breaks the balance and results in an incorrect solution. In our example, multiplying only the left side of the equation \(x+2=5\) by 2 led to a different and incorrect value for x. To keep the equation balanced, always perform the same operation on both sides of the equation.

Step 1: Example Equation

Consider the equation \(x + 2 = 5\).

Step 2: Multiply Just One Side

Suppose we decide to multiply only the left side of the equation by 2. Then, our equation would become \(2(x + 2) = 5\).

Step 3: Simplify the Equation

Now, let's simplify the equation we created by multiplying only the left side by 2. \(2(x + 2) = 2x + 4 = 5\)

Step 4: Attempt to Solve the Equation

Now, let's attempt to solve the equation for x: \(2x + 4 = 5\) Subtract 4 from both sides: \(2x = 1\) Divide by 2: \(x = 0.5\)

Step 5: Compare the Solutions

With our modified equation, we've determined x = 0.5. However, if we solve the original equation: \(x + 2 = 5\) Subtract 2 from both sides: \(x = 3\)

Step 6: Conclusion

In this example, you can see that when we multiplied just one side of the equation, it resulted in a different (and incorrect) value for x. Maintaining the balance of the equation is crucial for finding the correct algebraic solutions, which is why you must perform the same operation on both sides of an equation.

Key Concepts

Balance in Equations

In algebra, the concept of balance in equations is a fundamental principle. Think of an equation as resembling a balanced scale. This means that whatever operation you perform to one side, you must also do to the other side to keep it balanced. If you imagine an equation as two sides of a scale, whatever you add, subtract, multiply, or divide on one side, must be done on the other to maintain equilibrium.

The example given in the exercise illustrates this very well. When solving the equation \(x + 2 = 5\), both sides of the equation must 'weigh' the same, meaning they must have equal values. When we perform an operation only on one side, like multiplying just the left side by 2, the equation becomes unbalanced. This breaks the equality, leading to incorrect solutions.

If the scale is not balanced, it will tip over, signifying a false equation. A balanced equation ensures that the properties of equality hold, making it consistent and true.

Solving Equations

Solving equations is a systematic process in algebra that requires manipulations to isolate the unknown variable. The main goal is to find the value of the unknown that makes the equation true. Equations can often appear complex, but by following certain steps, you can simplify and solve them.

In the given example, solving \(x + 2 = 5\), involves moving all terms containing the unknown, \(x\), to one side of the equation. We start by subtracting 2 from both sides to maintain balance, giving us \(x = 3\).

It's essential to remember that maintaining balance while performing operations is key to solving equations. Whether adding, subtracting, multiplying, or dividing terms, each step in solving the equation must preserve the equality, ensuring that both sides of the equation remain equal.

Algebraic Manipulation

Algebraic manipulation is the art of rearranging and simplifying expressions and equations. It involves applying various algebraic rules to isolate the variable of interest. This skill is crucial in both solving equations and ensuring that mathematical expressions are simplified correctly.

In the exercise example, the mistake of multiplying only one side demonstrates how improper manipulation can lead to wrong conclusions. Proper algebraic manipulation involves the careful and balanced application of operations. For instance, if you multiply or divide any side by a number, you must apply the same operation to the other side.

This might include using distributive, associative, or commutative laws to reorganize terms, factorizing expressions, or using inverse operations to simplify expressions. Mastering algebraic manipulation helps in efficiently solving equations and understanding deeper mathematical concepts. It ensures that each step taken in transforming an equation preserves its integrity and balance.