How To Solve For 'b' When 'a' Is Known

Dec 10, 2025 by Alex Johnson 39 views

In the realm of mathematics, problems often involve understanding the relationships between different variables. One common scenario is when you're given an equation that links two variables, and you need to find the value of one variable when the other is provided. Today, we're going to tackle a specific example: $b = 3 - 2a$ . We'll work out the value of $b$ when $a=4$ . This type of problem is fundamental in algebra and helps build a strong foundation for more complex mathematical concepts. Understanding how to substitute values and perform simple arithmetic operations is crucial not just for passing math tests, but for everyday problem-solving and logical thinking. Let's dive into the process, breaking it down step-by-step so it's easy to follow, even if you're just starting out with algebra.

Understanding the Equation: $b = 3 - 2a$

Before we jump into solving, let's take a moment to understand what the equation $b = 3 - 2a$ is telling us. This is a linear equation, meaning it describes a straight line when graphed. The equation establishes a direct relationship between the variables $a$ and $b$ . Specifically, it states that the value of $b$ is determined by subtracting twice the value of $a$ from the number 3. The term ' $2a$ ' means '2 multiplied by $a$ '. So, whatever value $a$ holds, we first double it and then subtract that result from 3 to find the corresponding value of $b$ . This equation is like a rule or a formula that always holds true for any pair of $(a, b)$ that satisfies it.

Variable 'a': This is an independent variable. Its value can be anything, and it affects the value of $b$ . In this problem, we are given a specific value for $a$ , which is 4.
Variable 'b': This is a dependent variable. Its value depends on the value of $a$ . We need to find this value.
The '3': This is a constant. It's a fixed number that doesn't change.
The '-2': This is the coefficient of $a$ . It tells us how many times $a$ is multiplied, and the negative sign indicates subtraction or an inverse relationship.

Understanding these components is the first step to confidently solving the problem. It's like knowing the ingredients and the cooking instructions before you start baking a cake. The structure of the equation guides our actions.

The Substitution Step: Plugging in the Value of 'a'

The core of solving this type of algebraic problem lies in substitution. Substitution is the process of replacing a variable with its given numerical value. In our case, we are told that $a = 4$ . So, our first action is to go into the equation $b = 3 - 2a$ and replace every instance of ' $a$ ' with the number '4'.

Let's write it out:

Original equation: $b = 3 - 2a$

Substitute $a=4$ : $b = 3 - 2(4)$

Notice how we've placed the '4' in parentheses. This is a good practice, especially when dealing with negative numbers or more complex expressions, as it helps to avoid errors and clearly shows what is being multiplied. The parentheses here explicitly indicate that we need to calculate $2 imes 4$ as the next step.

This substitution step is crucial because it transforms an equation with two unknowns into an equation with only one unknown ( $b$ ), which we can then solve using basic arithmetic. It's the bridge that connects the given information ( $a=4$ ) to the unknown we want to find ( $b$ ).

Performing the Calculation: Order of Operations

Now that we have substituted the value of $a$ , we have the expression $b = 3 - 2(4)$ . The next critical step is to perform the arithmetic operations correctly. To do this, we must follow the order of operations. A commonly used acronym for the order of operations is PEMDAS or BODMAS:

Parentheses (or Brackets)
Exponents (or Orders)
Multiplication and Division (from left to right)
Addition and Subtraction (from left to right)

In our expression $b = 3 - 2(4)$ , we look for parentheses first. Inside the parentheses, there's just the number 4. The multiplication is indicated by the '2' right next to the parentheses, meaning $2 imes 4$ .

So, according to PEMDAS, multiplication comes before subtraction. We calculate $2 imes 4$ first:

$2 imes 4 = 8$

Now, we substitute this result back into our equation:

$b = 3 - 8$

We have now simplified the equation to a simple subtraction problem. The final step is to perform this subtraction:

$3 - 8$

When subtracting a larger number from a smaller number, the result will be negative. Think of it as starting at 3 on a number line and moving 8 units to the left. You'll end up at -5.

$3 - 8 = -5$

Therefore, the value of $b$ is -5.

Final Answer and Verification

We have successfully worked out the value of $b$ when $a=4$ using the equation $b = 3 - 2a$ . The steps involved were:

Understand the equation: $b = 3 - 2a$
Substitute the given value: Replace $a$ with 4, giving $b = 3 - 2(4)$ .
Apply order of operations (PEMDAS): First, calculate the multiplication $2 imes 4 = 8$ .
Perform the subtraction: $b = 3 - 8 = -5$ .

So, the final answer is $b = -5$ .

Verification: It's always a good idea to check your answer. We can do this by plugging our calculated value of $b$ (-5) and the given value of $a$ (4) back into the original equation to see if it holds true.

Equation: $b = 3 - 2a$

Substitute $a=4$ and $b=-5$ :

$-5 = 3 - 2(4)$

Calculate the right side:

$-5 = 3 - 8$

$-5 = -5$

Since both sides of the equation are equal, our solution is correct! This verification process confirms that when $a$ is 4, $b$ must indeed be -5 for the equation $b = 3 - 2a$ to be satisfied.

Understanding these basic algebraic manipulations is fundamental. Whether you're dealing with simple linear equations or more complex systems, the principles of substitution and following the order of operations remain constant. Keep practicing, and you'll find these types of problems become second nature. For further exploration into algebraic concepts, you might find resources on Khan Academy incredibly helpful, offering a wealth of information and practice exercises.