Problem 12 - Algebra 1: Concepts and Skills

Question

Solve the equation. \(-8+5 a-2=20\)

Step-by-Step Solution

Answer

The solution to the equation is \(a = 6\).

Step 1: Rearrange the equation

In the first step, the main objective is to place the terms involving the variable 'a' on one side of the equation and the constants on the other side of the equation. Here, the variable of interest is 'a'. To do this, add 2 to both sides of the equation: \(-8+5a-2+2=20+2\). This simplifies to \(5a-8=22\). Then, organise the terms, thus yielding a simple equation: \(5a = 22 + 8\).

Step 2: Simplify the equation

On the right-hand side, add 22 and 8 to yield \(5a = 30\).

Step 3: Solve for the variable

Now divide both sides by 5 so that 'a' is on one side of the equation by itself. This yields the solution: \( a = \frac{30}{5}\).

Key Concepts

Algebra

Algebra is a significant branch of mathematics that deals with symbols and the rules for manipulating these symbols. It provides a powerful framework for solving problems that involve unknown values, known as variables.

In algebra, we often work with equations to find these unknown values. An equation is a statement that equates two expressions. For example, in the equation \(-8 + 5a - 2 = 20\), the purpose is to find the value of 'a' that satisfies the equation. To do this, we rely on a fundamental property of equations: we can perform the same mathematical operation on both sides without changing the equation's balance. This allows us to isolate the variable and solve for it efficiently.

Understanding the foundational principles of how algebra works is crucial because it opens doors to more advanced math topics and practical applications in fields like physics, engineering, and computer science.

Variables

In mathematics, variables are symbols that represent numbers whose values are not yet known. They are a core concept in algebra, giving us the flexibility to refer to numbers without knowing their exact values.

• Variables are often represented by letters such as \(a\), \(x\), or \(y\).
• They allow us to create general formulas and express relationships between quantities.

In the equation \(-8 + 5a - 2 = 20\), 'a' is the variable. The goal is to determine the value of 'a' that makes the equation true. Variables can stand for different numbers in different situations, providing a dynamic tool for problem-solving.
Understanding how to manipulate variables is essential for developing analytical skills and critical thinking, which are applicable in various areas like science, engineering, and economics.

Linear Equations

Linear equations are algebraic equations in which each term is either a constant or the product of a constant and a single variable. These equations are usually in the form \(ax + b = c\). They can be solved to find the value of a variable. The equation used in our example is a linear equation because it involves a single variable, 'a', with no exponents or complex functions.
Solving such equations generally involves:

• Rearranging the equation to isolate the variable on one side.
• Performing operations such as addition, subtraction, multiplication, or division on both sides of the equation to solve for the variable.

Linear equations are foundational in algebra because they describe straight-line relationships between variables. This simplicity allows us to model and solve problems efficiently in a wide variety of disciplines, from business to the sciences. By getting comfortable with solving linear equations, you build a critical skill that underpins much of higher-level math.