Solving 4x-3=21 Step-by-Step And Graphing The Solution

Hey guys! Today, we're diving into the exciting world of algebra, where we'll be solving equations and visualizing their solutions on a number line. This is a fundamental concept in mathematics, and mastering it will open doors to more advanced topics. Let's break it down step-by-step and make sure we understand every detail.

Understanding the Equation

Our main focus is on the equation 4x - 3 = 21. In this equation, x represents an unknown value that we need to find. The goal is to isolate x on one side of the equation so that we can determine its value. To do this, we'll use inverse operations, which are operations that undo each other. For example, addition is the inverse operation of subtraction, and multiplication is the inverse operation of division.

The left side of the equation, 4x - 3, means that we're multiplying the unknown value x by 4 and then subtracting 3. The right side of the equation, 21, is a constant value. The equals sign (=) tells us that the expression on the left side has the same value as the number on the right side. Solving an equation involves finding the value of x that makes this statement true.

Before we jump into solving, it's essential to grasp the underlying principles. Think of an equation like a balanced scale. Whatever we do to one side, we must also do to the other side to maintain the balance. This is the golden rule of equation solving! If we add something to the left side, we need to add the same thing to the right side. If we multiply the right side by a number, we need to multiply the left side by the same number. Keeping the equation balanced ensures that we're correctly finding the value of x.

Step 1: Adding 3 to Both Sides

The first step in solving our equation 4x - 3 = 21 is to get rid of the -3 on the left side. To do this, we'll use the inverse operation of subtraction, which is addition. We'll add 3 to both sides of the equation. This maintains the balance and helps us isolate the term with x.

So, we add 3 to both sides:

4x - 3 + 3 = 21 + 3

On the left side, the -3 and +3 cancel each other out, leaving us with 4x. On the right side, 21 + 3 equals 24. So, our equation now looks like this:

4x = 24

This step is crucial because it simplifies the equation and brings us closer to isolating x. By adding 3 to both sides, we've eliminated the constant term on the left, making it easier to deal with the term involving x. Remember, the goal is to get x by itself on one side of the equation, and this step is a significant move in that direction.

Think of it like peeling back layers of an onion. We're gradually stripping away the constants and coefficients until we get to the core, which is x itself. Each step we take brings us closer to the final solution. This methodical approach is key to solving algebraic equations accurately and efficiently. The beauty of algebra lies in its systematic nature, where each operation serves a specific purpose in unraveling the unknown.

Step 2: Dividing Both Sides by 4

Now that we have 4x = 24, we need to isolate x completely. The 4 in 4x means 4 times x. To undo this multiplication, we'll use the inverse operation, which is division. We'll divide both sides of the equation by 4. Again, this keeps the equation balanced and helps us find the value of x.

So, we divide both sides by 4:

(4x) / 4 = 24 / 4

On the left side, the 4 in the numerator and the 4 in the denominator cancel each other out, leaving us with just x. On the right side, 24 divided by 4 is 6. So, we have:

x = 6

We've successfully solved for x! This means that the value of x that makes the original equation 4x - 3 = 21 true is 6. This is the solution to our equation. Dividing both sides by 4 was the final step in isolating x, and it allowed us to determine its value.

This step highlights the importance of understanding inverse operations. We used division to undo multiplication, just as we used addition to undo subtraction in the previous step. These inverse operations are the tools we use to manipulate equations and solve for unknowns. Each operation is like a key that unlocks a piece of the puzzle, bringing us closer to the solution. The process of solving equations is a journey, and each step is a milestone along the way.

Plotting the Solution on a Number Line

Now that we've found the solution x = 6, let's visualize it on a number line. A number line is a simple but powerful tool that helps us understand the position and relationship of numbers.

To plot x = 6 on a number line, we first draw a straight line. Then, we mark a point on the line and label it as 0. This is our reference point, also known as the origin. Next, we mark other points on the line at equal intervals, representing integers (whole numbers). We can label these points as 1, 2, 3, 4, 5, 6, and so on, to the right of 0. To the left of 0, we have negative integers: -1, -2, -3, and so on.

To plot x = 6, we find the point on the number line that corresponds to the number 6. Since 6 is a positive number, it will be to the right of 0. We place a dot or a filled circle at the point representing 6. This dot visually represents the solution to our equation.

Plotting solutions on a number line is a great way to reinforce our understanding of what the solution means. It's not just an abstract number; it's a specific point on a line that represents a value. This visual representation can be particularly helpful when dealing with inequalities, where the solution might be a range of values rather than a single point. In those cases, we would shade a portion of the number line to represent the solution set.

The number line is a fundamental tool in mathematics, and it's used in various contexts, from basic arithmetic to more advanced topics like calculus. It helps us visualize numbers, their order, and the relationships between them. Mastering the number line is an essential step in building a solid foundation in mathematics.

Summary Table

To summarize our steps, here's a table:

This table neatly organizes the steps we took to solve the equation. It shows the original equation, the operations we performed, and the resulting equations. This can be a helpful way to review the process and make sure we understand each step.

Creating tables like this is a great habit to develop when solving math problems. It helps us stay organized, track our progress, and avoid mistakes. It's also a useful tool for communicating our solution process to others. A well-organized table can make a complex problem much easier to understand.

Conclusion

So, there you have it! We've successfully solved the equation 4x - 3 = 21 and plotted the solution x = 6 on a number line. Remember, the key is to use inverse operations to isolate the variable and keep the equation balanced. This is a fundamental concept in algebra, and practicing these steps will make you a pro at solving equations. Keep practicing, guys, and you'll master it in no time!

This example demonstrates the core principles of solving linear equations. These principles can be applied to a wide range of problems, and mastering them is crucial for success in algebra and beyond. The ability to solve equations is not just a mathematical skill; it's a problem-solving skill that can be applied to many areas of life. So, keep honing your skills, and you'll be well-equipped to tackle any challenge that comes your way.