Problem 83 - Algebra 1

Question

Solve the equation. $$ 3 p-12=6 $$

Step-by-Step Solution

Answer

The solution to the equation \(3p - 12 = 6\) is \(p = 6\).

Step 1: Identify the equation

The given equation is \(3p - 12 = 6\). The task is to isolate the variable \(p\).

Step 2: Shift the constant term

Add \(12\) to both sides of the equation to eliminate the '-12' on the left side of the equation: \(3p = 6 + 12\)

Step 3: Simplify the right side

Simplify the right side of the equation: \(3p = 18\)

Step 4: Isolate the p term

Divide both sides by 3 to isolate p: \(p = 18/3\)

Step 5: Simplify the quotient

Upon simplifying the right side we find the solution: \(p = 6\)

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 a single variable. Linear equations occur abundantly in various fields such as physics, economics, and engineering, making them crucial to understand. The general form of a linear equation in one variable is expressed as \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable we aim to solve for.

Linear equations have key properties such as a straight-line graph representation and a single solution for the variable, which makes them particularly appealing for solving real-world problems. They strike a balance between simplicity and utility, providing a foundational tool for more complex mathematical concepts.

Isolate Variable

To isolate the variable means to manipulate an equation in such a way that the variable we want to solve for stands alone on one side of the equation, and everything else is on the opposite side. This process is essential for solving equations, as it gives us the value of the variable in question.

• Add or subtract terms to get the variable term on one side and the constants on the other.
• Use multiplication or division to get rid of any coefficients attached to the variable.

By following these steps, we reach the simplest form of the equation which directly reveals the value of the variable. For instance, in the equation \(3p - 12 = 6\), to isolate \(p\), we would first get rid of the constant \(\-12\) and then deal with the coefficient \(3\).

Equation Simplification

Equation simplification is the process of reducing an equation to its simplest form, making the value of the unknown variable clear and easy to understand. Simplification can involve:

• Combining like terms, which are terms that contain the same variables to the same power,
• Cancelling out terms by adding or subtracting them from both sides of an equation,
• Rearranging the terms for a clearer view of the relationship between variables and constants.

As in our provided example, once we moved the constant term to the other side to get \(3p = 18\), we simplified the equation by dividing both sides by \(3\), which resulted in \(p = 6\). This final simplification allowed us to clearly see the value of \(p\).