Solving Equations: Is X A Solution?

Dec 4, 2025 by Andrew McMorgan 36 views

Hey guys! Ever wonder if a number is the real deal when it comes to solving an equation? Let's break it down and see if we can make sense of it all. We're going to take a look at the equation x - 39 = 10 and figure out whether the given values for x (which are 46 and 65) actually make the equation true. It's like a detective game, and we're the detectives!

What Does it Mean to Be a Solution?

Okay, so what does it even mean for a number to be a solution to an equation? In simple terms, a solution is a value that, when you plug it in for the variable (in our case, x), makes the equation a true statement. Think of it like this: the equation is a question, and the solution is the correct answer. If you substitute the solution into the equation, the left side of the equation will be equal to the right side. If it doesn't, then the number is not a solution.

For example, if we had the equation x + 2 = 5, the solution would be x = 3 because 3 + 2 = 5. Easy peasy, right? We are using this same process to find out if x = 46 and x = 65 are solutions. In this first section, we want to define clearly what a solution means. To be a solution you must fulfill the requirements of the equation provided.

Let's continue breaking this down further, when we're dealing with equations, the ultimate goal is often to find the value(s) of the variable(s) that make the equation true. These values are called solutions. So, when we ask if a certain value of x is a solution to x - 39 = 10, we're essentially asking: "Does plugging in this value for x make the equation a true statement?"

Now, how do we check if a value is a solution? Simple! We substitute the value for the variable in the equation and see if the equation holds true. In other words, we perform the arithmetic on the left-hand side (LHS) of the equation and see if it equals the right-hand side (RHS). If LHS = RHS, then the value is a solution. If LHS ≠ RHS, then the value is not a solution.

Testing x = 46

Alright, let's get our hands dirty and test the first value, x = 46. We'll substitute 46 for x in the equation x - 39 = 10. So, we have:

46 - 39 = 10

Now, let's simplify the left side:

46 - 39 = 7

So, the equation becomes:

7 = 10

Is this a true statement? Nope! 7 is definitely not equal to 10. Therefore, x = 46 is not a solution to the equation x - 39 = 10. It's like trying to fit a square peg into a round hole – it just doesn't work! So, when x is 46, it does not satisfy the equation x - 39 = 10.

When testing if x = 46, we substitute the value into the original equation. This is the fundamental concept of evaluating whether a given value satisfies an equation. When we substitute x = 46 into the equation x - 39 = 10, we get 46 - 39 = 10. Evaluating the left-hand side, we find that 46 - 39 = 7. Thus, the equation becomes 7 = 10, which is clearly false.

Since 7 ≠ 10, we conclude that x = 46 is not a solution to the equation x - 39 = 10. This demonstrates the importance of substituting the given value into the original equation and checking if the resulting statement is true. If the statement is false, then the given value is not a solution. In this case, x = 46 does not satisfy the equation, so it is not a valid solution.

This process highlights the basic principle of verifying solutions in algebraic equations. By substituting a potential solution into the equation and checking if the resulting statement is true, we can determine whether the value is a valid solution. This method is widely used in algebra to solve and verify equations.

Testing x = 65

Okay, let's move on to the next value, x = 65. We'll do the same thing: substitute 65 for x in the equation x - 39 = 10:

65 - 39 = 10

Now, let's simplify the left side:

65 - 39 = 26

So, the equation becomes:

26 = 10

Is this a true statement? Nope again! 26 is definitely not equal to 10. Therefore, x = 65 is also not a solution to the equation x - 39 = 10. It seems like we're on a roll with non-solutions! These are very important skills when solving equations. We have determined that when x = 65, it does not satisfy the equation x - 39 = 10.

Now, let's move on to the next value, x = 65. We'll do the same thing: substitute 65 for x in the equation x - 39 = 10. So, we have: 65 - 39 = 10. Now, let's simplify the left side: 65 - 39 = 26. So, the equation becomes: 26 = 10. Is this a true statement? Nope again! 26 is definitely not equal to 10. Therefore, x = 65 is also not a solution to the equation x - 39 = 10.

In this case, when testing if x = 65, we substitute the value into the original equation. This is the fundamental concept of evaluating whether a given value satisfies an equation. When we substitute x = 65 into the equation x - 39 = 10, we get 65 - 39 = 10. Evaluating the left-hand side, we find that 65 - 39 = 26. Thus, the equation becomes 26 = 10, which is clearly false.

Since 26 ≠ 10, we conclude that x = 65 is not a solution to the equation x - 39 = 10. This reinforces the importance of substituting the given value into the original equation and checking if the resulting statement is true. If the statement is false, then the given value is not a solution. In this case, x = 65 does not satisfy the equation, so it is not a valid solution.

Finding the Real Solution

So, neither 46 nor 65 are the solutions. What is the real solution then? To find the correct solution, we need to isolate x on one side of the equation. We can do this by adding 39 to both sides of the equation:

x - 39 + 39 = 10 + 39

This simplifies to:

x = 49

Aha! The real solution is x = 49. If we substitute 49 back into the original equation, we get:

49 - 39 = 10

10 = 10

That's a true statement! So, x = 49 is indeed the solution.

To find the real solution to the equation x - 39 = 10, we need to isolate x on one side of the equation. We can do this by adding 39 to both sides of the equation: x - 39 + 39 = 10 + 39. This simplifies to x = 49. Now, let's check if x = 49 is indeed the solution. We substitute 49 back into the original equation: 49 - 39 = 10. This simplifies to 10 = 10, which is a true statement. Therefore, x = 49 is indeed the solution to the equation x - 39 = 10.

This process demonstrates how to solve a simple linear equation by isolating the variable. By adding the same value to both sides of the equation, we maintain the equality and can isolate x to find its value. This is a fundamental technique in algebra and is used to solve a wide range of equations.

Key Takeaways

A solution to an equation is a value that makes the equation true when substituted for the variable.
To check if a value is a solution, substitute it into the equation and see if the left side equals the right side.
If the equation is not true after substituting, then the value is not a solution.
To find the actual solution, isolate the variable on one side of the equation using algebraic operations.

So there you have it, guys! Now you know how to determine whether a value is a solution to an equation. Keep practicing, and you'll become equation-solving pros in no time!

Practice Problems:

Is x = 5 a solution to 2x + 3 = 13?
Is x = -2 a solution to x - 4 = -6?

I hope this helps. Keep up the excellent work.