Problem 11 - Elementary Algebra Within Reach

Question

Decide which operation you would use first to solve the equation. $$3 x-6=0$$

Step-by-Step Solution

Answer

To solve the equation \(3x - 6 = 0\), addition would be used first, followed by division. \(x = 2\).

Step 1: Add or subtract to isolate term with \(x\)

Add 6 to both sides of the equation to isolate the term with \(x\): \(3x - 6 + 6 = 0 + 6\). This results in the equation: \(3x = 6\).

Step 2: Divide to isolate \(x\)

Divide both sides of the new equation by 3 to solve for \(x\): \(3x / 3 = 6 / 3\). This simplifies to \(x = 2\).

Key Concepts

Isolation of Variable

In solving linear equations, the goal is to find the value of the variable that makes the equation true. The first step is often to isolate the variable. This means we move all instances of the variable to one side of the equation. We do this by performing operations that reverse what is done in the equation. If something is added, we subtract it. If it's multiplied, we divide it. The purpose is to have the variable stand alone, making it easier to find its value.For example, consider the equation in our exercise, which is \(3x - 6 = 0\). Our task is to isolate \(x\) on one side. To begin, we need to get rid of \(-6\), which is on the same side as \(x\). By adding 6 to both sides of the equation, we move the \(-6\) away, simplifying the equation to \(3x = 6\). Now, the term with \(x\) is isolated on one side.

Addition and Subtraction in Equations

When dealing with linear equations, one of the most common techniques is using addition and subtraction to manipulate the terms. These operations help shift terms from one side of the equation to the other, assisting in isolating the variable.

• Addition cancels out subtraction: If a term is subtracted, adding the same value will result in zero.
• Subtraction cancels out addition: Similarly, if a term is added, subtracting the same value will eliminate it from that side of the equation.

Continuing with our exercise, in the equation \(3x - 6 = 0\), the \(-6\) was subtracted from \(3x\). Therefore, we perform the inverse operation, which is addition. By adding 6 to both sides—a crucial step to maintain the balance of equality—we eliminate \(-6\) on the left, transforming the equation into \(3x = 6\). This application of addition or subtraction allows us to manage and rearrange terms efficiently to progress toward isolating the variable.

Division in Equations

Dividing in equations is another fundamental operation that is utilized especially after using addition or subtraction. Once the variable term is isolated, you might need to perform division to ensure the variable is truly isolated, ultimately solving the equation.For instance, from the equation derived in our exercise, \(3x = 6\), \(x\) must be isolated. Since \(x\) is currently multiplied by 3, we need to use division to "cancel out" this multiplication, leaving \(x\) alone. The operation involves the following steps:

• Divide every term in the equation by the same non-zero number—in this case, 3, because that is what we multiplied \(x\) by.
• This leads us to \(x = \frac{6}{3}\).
• Simplifying gives us \(x = 2\).

It is crucial to apply the division to each side of the equation to keep it balanced. This approach of using division helps in extracting the solution by transforming complex equations into simpler, easily interpretable results.