Question

(a) Write down the general form of a linear equation. (b) Explain what is meant by the root of a linear equation.

Short answer

Question: Explain the general form of a linear equation and the meaning of its root. Provide an example to demonstrate how to find the root of a linear equation. Answer: The general form of a linear equation is given by \(ax + b = 0\), where \(a\) and \(b\) are constants, \(x\) is the variable, and \(a\) cannot be equal to zero. The root of a linear equation is the value of the variable for which the equation is true or satisfied, or the value of \(x\) that makes the left side of the equation equal to the right side. For example, consider the linear equation \(2x - 4 = 0\). To find the root, we need to find the value of \(x\) that satisfies the equation. We can do this by solving the equation for \(x\): 1. Add 4 to both sides of the equation: \(2x = 4\) 2. Divide both sides by 2: \(x = 2\) Since the equation is true when \(x = 2\), the root of the given linear equation is 2.

Step-by-step solution

Define a linear equation

A linear equation is an algebraic equation in which each term is either a constant or a product of a constant and a single variable. It has only one degree, which means that the highest exponent of the variable is 1.

Write the general form of a linear equation

The general form of a linear equation is given by: \(ax + b = 0\), where \(a\) and \(b\) are constants, \(x\) is the variable, and \(a\) cannot be equal to zero. If \(a\) were zero, the equation would not be linear but a constant equation.

Define the root of a linear equation

The root of a linear equation is the value of the variable for which the equation is true or satisfied. In other words, it is the value of \(x\) that makes the left side of the equation equal to the right side.

Explain the root with an example

Let's consider a linear equation: \(2x - 4 = 0\). To find the root, we need to find the value of \(x\) that satisfies the equation. We can do this by solving the equation for \(x\). 1. Add 4 to both sides of the equation: \(2x = 4\) 2. Divide both sides by 2: \(x = 2\) Since the equation is true when \(x = 2\), the root of the given linear equation is 2.

Key concepts

Algebraic Equations

Algebraic equations are the cornerstone of algebra and are fundamental to understanding mathematical and scientific concepts. They are a form of mathematical statement that uses numerical and literal (using letters) symbols to express a relationship between values. Algebraic equations can vary in complexity from simple linear equations to more complex quadratic equations or polynomials.

In the simplest form, an algebraic equation is made up of two expressions set equal to each other, consisting of variables, coefficients, and constants. For example, the equation \(3x + 5 = 11\) includes the variable \(x\), the coefficients 3 and 1 (where 1 is the coefficient of 5, since any number can be written as itself multiplied by 1), and the constant 5.

These mathematical constructs allow individuals to find unknown values, known as variables, by using operations that include addition, subtraction, multiplication, and division. Understanding how to manipulate these equations is essential for working through more advanced areas of mathematics and various applications in science and engineering.

Root of a Linear Equation

The root of a linear equation, often also referred to as the solution or zero of the equation, is a fundamental concept in algebra. It's the exact value that, when substituted into the equation for the variable, will satisfy the equation and make it true. In the context of the equation \(ax + b = 0\), the root is the value of \(x\) that results in a zero value for the entire expression.

Understanding the significance of a root goes beyond just solving for \(x\); it has a graphical interpretation as well. On a graph, the root of a linear equation represents the point where the line crosses the \(x\)-axis. This is because the \(y\) value for any point on the \(x\)-axis is zero, which aligns with the concept of the linear equation's root leading to a zero value. Therefore, solving a linear equation not only provides the roots numerically but also gives insight into the linear relationship's graphical behavior.

Solving Linear Equations

Solving linear equations is a foundational skill in algebra that involves finding the value(s) of the variable(s) that satisfy the equation. To solve a linear equation such as \(ax + b = 0\), the goal is to isolate the variable \(x\) on one side of the equation. This process typically involves several steps:

• Using inverse operations to move terms from one side of the equation to the other.
• Simplifying each side of the equation if necessary by combining like terms or factoring.
• Checking the solution by substituting it back into the original equation to ensure it results in a true statement.

With the equation \(2x - 4 = 0\) as an example, the steps are adding 4 to both sides to get \(2x = 4\) and then dividing both sides by 2 to find \(x = 2\), which is the root of the equation.

It's important to understand these steps because they are not just for solving a single linear equation but can be applied to systems of linear equations and inequalities as well. Efficiency and accuracy in this process are crucial, as is the ability to apply these skills to real-world problems in various fields such as physics, economics, and computer science.