Solving Equations: Substitution Method Explained

Hey Plastik Magazine readers! Let's dive into a cool math concept: solving equations using the substitution method. It might sound a bit intimidating at first, but trust me, it's pretty straightforward, and we'll break it down step by step. We're going to use the set {$9, 11, 14$} and the equation $87 - 4x = 43$ to find the value of x that makes the equation true. Let's get started, shall we?

Understanding the Substitution Method

So, what exactly is the substitution method? Think of it like a detective game, guys. We have an equation, and our goal is to find the secret value of x that, when plugged into the equation, makes everything balance out perfectly. With the substitution method, we're given a set of possible values for x. We take each of these values, one by one, and 'substitute' them into the equation in place of x. Then, we do the math to see if the equation holds true. If it does, voilà! We've found our solution. If it doesn't, we move on to the next possible value. This method is especially helpful when we're given a limited set of options, as it simplifies the process of finding the right answer. We're essentially testing each value to see which one 'fits' the equation.

It's like trying different keys to unlock a door. You try one key, and if it opens the door, you've found the right one. If it doesn't, you try another until you find the perfect match. This method is a direct and hands-on approach, great for understanding how the variables work within an equation. The process is clear: replace the variable, simplify the expression, and evaluate the truth of the equation. This makes it an ideal strategy for beginners in algebra, as it reinforces the understanding of how numbers work together. We're focusing on a basic algebraic principle today, but remember that the applications of this method are widespread throughout mathematics and beyond. This approach is not only useful for solving equations, but also for understanding functions, graphs, and complex mathematical models. By mastering the fundamentals, you equip yourself with the tools to solve more intricate problems down the road. This method provides a clear visual representation of what the solutions mean: it means that by substituting the value of x into the equation, you can get a true statement. It's a very practical way to learn about algebra. Let's get our hands dirty and start substituting!

Applying Substitution to Our Equation

Alright, let's get down to business with our equation: $87 - 4x = 43$. We're given the set {$9, 11, 14$}, which means our possible values for x are 9, 11, and 14. We'll take each of these numbers, one at a time, and plug them into the equation to see if they satisfy it. Remember, our aim is to find which number, when substituted for x, makes the left side of the equation equal to the right side (43). So, let's start with the first value, which is 9. We're going to substitute 9 for x in the equation.

We'll then solve the equation with 9 as the value of x. The equation becomes $87 - 4 * 9 = 43$. Let's start the evaluation, so $87 - 36 = 43$. It's not the same; so, $51 = 43$. The equation isn't true for x = 9. So, we'll try the next value in our set. This means that x = 9 is not the solution to the equation. We move to the next possible value, which is 11, substituting it into our original equation. We'll then evaluate the equation with 11 as the value of x. The equation turns into $87 - 4 * 11 = 43$. Doing the math, we have $87 - 44 = 43$. It's not the same; so, $43 = 43$. The equation is true for x = 11. Since we found a value that satisfies the equation, we can stop here. We don't need to try the last value, which is 14. We've found our answer, guys! The value of x that makes the equation true is 11.

Step-by-Step Breakdown

Okay, let's break this down into clear, easy-to-follow steps to make sure we've got it down pat. This step-by-step approach will help solidify your understanding and make future problem-solving a breeze.

• Step 1: Understand the Equation. Make sure you know what the equation is and what the variable (x, in our case) represents. Our equation is $87 - 4x = 43$, and we're trying to find the value of x that makes it true. Remember that x is a number that we need to find, and when we multiply it by -4, and add 87, it equals 43.
• Step 2: Identify the Possible Values. We're given the set {$9, 11, 14$}, which gives us our possible values for x. This limits the number of options we need to test, which makes the problem simpler. Always look for this set, since they are the ones we will use to test our equation. It is also important to remember the order.
• Step 3: Substitute and Calculate. Start with the first value (9) and substitute it into the equation. So, the equation becomes $87 - 4 * 9 = 43$. Simplify the left side of the equation. We do this by following the order of operations, so multiplication first: $87 - 36 = 43$. Then, complete the subtraction to get $51 = 43$. Since this is not true, 9 is not the solution. Next, try the second value (11) and substitute it into the equation: $87 - 4 * 11 = 43$. Simplify: $87 - 44 = 43$, so $43 = 43$. This is true! We found our solution.
• Step 4: Verify and Conclude. Since the equation holds true when x = 11, we know that 11 is the solution. We don't need to try 14. We have solved the equation using substitution. Always double-check your work to avoid silly mistakes. Be mindful of the signs of the numbers.

Why Substitution Matters

You might be thinking,