91Ó°ÊÓ

In the world of linear equations, understanding the concept of a "root" is essential. A root of an equation is a value that makes the equation true when substituted into it. For instance, if you have a linear equation like \(3x + b = 0\), the root is any value of \(x\) that satisfies the equation.
Let's say we're given that \(x = -2\) is a root. This tells us that when \(-2\) is plugged into the equation, it will hold true. This means:

• Substitute \(x = -2\) into the equation.
• The resulting expression should equal zero.

In our example, plugging \(-2\) into \(3x + b = 0\) gives us a new equation \(3(-2) + b = 0\). This equation confirms that \(-2\) is indeed a root because it makes the left-hand side equal the right-hand side (zero). This concept is critical as it provides a tool for checking solutions and verifying their correctness.

Algebraic substitution is a technique used to simplify equations and expressions. It involves replacing a variable with a given value or another expression. When we know that \(x = -2\) is a root of the equation \(3x + b = 0\), substitution becomes very useful.
Here's how it works:

• Identify what you need to substitute; in this case, it's \(x = -2\).
• Replace \(x\) with \(-2\) in the equation, leading to \(3(-2) + b = 0\).
• Now, the problem becomes a simple arithmetic puzzle.

This substitution transforms the problem into something manageable. You are now dealing with concrete numbers instead of abstract variables, which makes it easier to simplify and solve the equation. It’s an invaluable tool for breaking down more complicated problems and should be used whenever you have information about the values of your variables.

The ultimate goal in many algebraic exercises is solving for the unknown value in an equation. Once you substitute and simplify, you often reach an equation that requires isolating the unknown. In our exercise, after applying the substitution, the equation simplifies to \(-6 + b = 0\).
To solve for \(b\):

• Look to isolate \(b\) on one side of the equation.
• In \(-6 + b = 0\), you can add 6 to both sides, resulting in \(b = 6\).
• This gives you the value of \(b\) that satisfies the equation when \(x = -2\).

Solving for unknowns is fundamentally about isolating one variable and calculating its value. By using processes like addition or subtraction, you can transform what might seem complicated into something straightforward. Mastering this process will enable you to tackle a wide range of algebraic equations with confidence.