Problem 71 - Elementary Algebra Within Reach

Question

Solve the equation and check your solution. $$15 x-3=15-3 x$$

Step-by-Step Solution

Answer

The solution of the given equation, \(15x - 3 = 15 - 3x\), is \(x = 1\).

Step 1: Group like terms

First, group like terms by adding \(3x\) to both sides of the equation to cancel \(3x\) on the right side. The equation becomes \(15x + 3x = 15 + 3\).

Step 2: Simplify both sides

After grouping like terms, simplify both sides of the equation. The equation now becomes \(18x = 18\).

Step 3: Solve for the variable

To find the value of 'x', divide both sides of the equation by 18. This gives \(x = 18/18\).

Step 4: Check the solution

To verify the solution, substitute \(x = 1\) into the original equation and confirm that both sides of the equation are equal. Substituting the value of \(x\) into the original equation results in \(15(1) - 3 = 15 - 3(1)\), which simplifies to \(12 = 12\), and confirms that the solution is correct.

Key Concepts

Understanding Like Terms

When solving equations, like terms are your work's friendly helpers. Like terms are terms in an equation that have identical variable parts. In our example, the like terms are those that include the variable \(x\). They are \(15x\) and \(-3x\).

Identifying like terms makes the process of solving equations smoother. You can add or subtract them, which is exactly what happens in the first step of the solution above. This means:

• If you have \(ax\) and \(bx\), you can combine them to become \((a+b)x\).

When we move \(3x\) from the right side to the left side of our equation, we deal with like terms efficiently. This leads to an equivalent equation where similar items are grouped together. By doing so, the equation starts to look simpler and more straightforward!

Simplifying Equations: The Key to Clarity

Equation solving is like peeling an onion; you simplify layer by layer. After grouping like terms, the next step is to simplify further. Simplifying the equation helps in making it less complex and easier to solve. Let’s take a closer look at the equation from our example:

Once the like terms are combined, the equation becomes \(18x = 18\). At this stage, simplifying means ensuring each side of the equation is as reduced as possible. This often involves combining numbers or terms. Here, it’s about recognizing that we have one term on each side:

• On the left: a single term \(18x\).
• On the right: a single term, 18.

This leads to an ideal scenario for solving equations, where fewer terms make it easier for you to find what the variable should be. Remember, a simplified equation is less prone to errors and easier to manage!

Dividing Both Sides of the Equation: Solving for the Variable

After simplifying the equation, it's time for the crucial move: solving for the variable. This usually involves isolating the variable, such as \(x\), on one side of the equation. In our situation, we had the simplified equation \(18x = 18\).

The goal is to have \(x\) by itself on one side. To achieve this, you divide both sides of the equation by the coefficient of \(x\) — which is 18. Let’s see why this step is necessary:

• By dividing both sides equally, you maintain the balance of the equation.
• This operation simplifies \(18x\) to \(x\), as \(\frac{18x}{18} = x\).
• On the right side, \(\frac{18}{18} = 1\), simplifying to just 1.

Hence, \(x = 1\). Dividing both sides truly cherishes the equality of the equation allowing an unobstructed view of the variable’s value.