Problem 47 - Elementary Algebra Within Reach

Question

Solve the equation and check your solution. $$5(3-x)=x-12$$

Step-by-Step Solution

Answer

The solution for the equation is \( x = 4.5 \).

Step 1: Distribute

First, distribute the 5 on the left side of the equation. So \(5 \cdot (3-x)\) becomes \(15 - 5x\) and we get \(15 - 5x = x - 12\).

Step 2: Simplify

Next, simplify by adding 5x to both sides of the equation. This results in \(15 = 6x - 12\).

Step 3: Isolate the variable

Proceed to isolate the variable x by adding 12 to both sides of the equation. The equation becomes \(27 = 6x\).

Step 4: Solve for x

Finally, to solve for x, divide both sides of the equation by 6 and get \( x = \frac{27}{6} = 4.5\).

Step 5: Check the solution

To confirm the solution, plug the answer back into the original equation \(5(3-x)=x-12\). So, \(5(3-4.5)= 4.5 -12\), which simplifies to \(-7.5= -7.5\). Because both sides of the equation are equal, the solution is valid.

Key Concepts

Linear Equation

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and the first power of a variable. These equations are called 'linear' because they represent straight lines in Cartesian two-dimensional space when you plot them on a graph. The standard form for a linear equation in one variable is:

\[\begin{equation} ax + b = 0 \text{, where } a \text{ and } b \text{ are constants.} \end{equation}\]

For instance, in the given exercise we have an equation with a simple format that makes variable isolation straightforward. The beauty of linear equations lies in their simplicity and the fact that they are the building blocks for understanding more complex algebraic concepts.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like x or y), and operators (like add, subtract, multiply, and divide). When an algebraic expression is set equal to another algebraic expression, it forms an equation. For example, in the exercise,

\[\begin{equation} 5(3-x) \text{ is an algebraic expression, and setting it equal to } x-12 \text{ gives us an equation to solve.} \end{equation}\]

Understanding how to manipulate these expressions using distributive, associative, and commutative properties is essential in simplifying and solving equations.

Variable Isolation

The process of variable isolation involves manipulating an equation to get the variable by itself on one side of the equal sign. This is a critical step in solving linear equations, as it helps us find the value of the variable, which is the solution to the equation. Here are the key concepts of variable isolation:

• Maintain Balance: Whatever you do to one side of the equation, do to the other.
• Reverse Operations: Use opposite operations to cancel out terms and isolate the variable.
• Work in Reverse Order of Operations: Try to undo addition or subtraction first, then multiplication or division.

In our exercise, isolating the variable meant getting x alone on one side, which was achieved through a series of strategic steps that simplified and reduced the equation to the simplest form where x was clearly defined.

Equation Checking

Equation checking is the final step in the problem-solving process to ensure that the value obtained for the variable indeed satisfies the original equation. This verification step is crucial; it's like proofreading your work before you submit it. Here's how you can check your solution:

1. Substitute the value back into the original equation.
2. Simplify the equation using the correct order of operations.
3. Ensure that both sides of the equation equal the same value.

In our solved exercise, we confirmed that x = 4.5 was correct by plugging it back into the equation and verifying that both sides were equal. This check is vital as it confirms the accuracy of your solution and reinforces your understanding of the algebraic principles involved.