Let's dive into solving this equation, guys! It looks a bit tricky at first glance, but we'll break it down step by step to make it super easy to understand. Our main goal here is to find the value of 'x' that satisfies the equation 9x + 6 = 2x + 4 = 10. This involves a bit of algebraic manipulation, but don't worry, we've got this!

Understanding the Equation

First, let's clarify what the equation 9x + 6 = 2x + 4 = 10 actually means. It seems like we have two separate equations combined into one. We need to treat them separately to find a consistent solution for 'x.' So, we can break it down into two equations:

1. 9x + 6 = 10
2. 2x + 4 = 10

Now that we have these two separate equations, we can solve each one individually and see if we get a consistent value for 'x.' If the 'x' values are different, it means there is no single solution that satisfies the entire original equation. Keep this in mind as we move forward.

Solving the First Equation: 9x + 6 = 10

Okay, let's tackle the first equation: 9x + 6 = 10. To solve for 'x,' we need to isolate 'x' on one side of the equation. Here’s how we do it:

1. Subtract 6 from both sides: This will get rid of the '+ 6' on the left side. So, we have: 9x + 6 - 6 = 10 - 6 9x = 4

2. Divide both sides by 9: This isolates 'x' completely. We get: 9x / 9 = 4 / 9 x = 4/9

So, for the first equation, we find that x = 4/9. Make a note of this value, as we'll compare it with the solution from the second equation.

Solving the Second Equation: 2x + 4 = 10

Now, let's move on to the second equation: 2x + 4 = 10. Again, our goal is to isolate 'x'. Here’s the breakdown:

1. Subtract 4 from both sides: This eliminates the '+ 4' on the left side: 2x + 4 - 4 = 10 - 4 2x = 6

2. Divide both sides by 2: This isolates 'x': 2x / 2 = 6 / 2 x = 3

So, for the second equation, we find that x = 3. Now we have two different values for 'x': 4/9 from the first equation and 3 from the second equation. This is a crucial point to observe.

Comparing the Solutions

Alright, guys, we've found that for the equation 9x + 6 = 10, x = 4/9, and for the equation 2x + 4 = 10, x = 3. Since these values are different, it means that there is no single value of 'x' that can satisfy the entire original equation 9x + 6 = 2x + 4 = 10. The equation, as it’s presented, has no consistent solution.

Why No Consistent Solution?

The problem arises because the equation implies that 9x + 6 and 2x + 4 must both equal 10 simultaneously. However, the values of 'x' that make each part of the equation true are different. This inconsistency means the original equation doesn't have a valid solution.

Checking Our Work

It's always a good idea to double-check our work to make sure we haven't made any mistakes. Let's plug our 'x' values back into the original equations to confirm:

• For x = 4/9:

  • 9(4/9) + 6 = 4 + 6 = 10 (This checks out)
  • 2(4/9) + 4 = 8/9 + 4 = 8/9 + 36/9 = 44/9 ≠ 10 (This does not check out)

• For x = 3:

  • 9(3) + 6 = 27 + 6 = 33 ≠ 10 (This does not check out)
  • 2(3) + 4 = 6 + 4 = 10 (This checks out)

As you can see, neither value of 'x' satisfies both parts of the original equation simultaneously. This confirms our conclusion that there is no consistent solution.

Conclusion

In conclusion, the equation 9x + 6 = 2x + 4 = 10 does not have a solution. We arrived at this conclusion by breaking down the equation into two separate equations, solving each one for 'x,' and then comparing the resulting 'x' values. Since the 'x' values were different, we determined that there is no single value of 'x' that can satisfy the entire original equation. Always remember to check your work and ensure that your solutions make sense within the context of the problem. Great job working through this problem with me, guys! Understanding these steps is crucial for tackling more complex algebraic equations in the future.

Additional Tips for Solving Equations

• Simplify: Always try to simplify the equation as much as possible before attempting to solve it. This might involve combining like terms or using the distributive property.
• Isolate the Variable: The main goal is to isolate the variable you are trying to solve for. Use inverse operations (addition, subtraction, multiplication, division) to get the variable by itself on one side of the equation.
• Check Your Work: After finding a solution, plug it back into the original equation to make sure it is correct. This can help you catch any mistakes you might have made along the way.
• Understand the Properties of Equality: These properties allow you to perform the same operation on both sides of the equation without changing its balance. For example, you can add the same number to both sides, or multiply both sides by the same number.
• Practice Regularly: The more you practice solving equations, the better you will become at it. Start with simple equations and gradually work your way up to more complex ones.

Real-World Applications of Solving Equations

Solving equations isn't just an abstract mathematical exercise; it has many practical applications in various fields. Here are a few examples:

• Engineering: Engineers use equations to design structures, circuits, and systems. They need to solve equations to determine the optimal dimensions, materials, and configurations.
• Physics: Physicists use equations to describe the behavior of the physical world. Solving these equations allows them to make predictions about everything from the motion of objects to the properties of light.
• Economics: Economists use equations to model economic phenomena, such as supply and demand, inflation, and economic growth. Solving these equations helps them understand and forecast economic trends.
• Computer Science: Computer scientists use equations to develop algorithms and software. Solving equations is essential for tasks such as optimization, data analysis, and machine learning.
• Finance: Financial analysts use equations to evaluate investments, manage risk, and make financial projections. Solving equations is crucial for tasks such as calculating interest rates, determining loan payments, and valuing assets.

By mastering the art of solving equations, you'll not only excel in mathematics but also gain valuable skills that can be applied in numerous real-world scenarios. So, keep practicing, stay curious, and embrace the power of equations!

Further Practice Problems

To solidify your understanding, here are a few practice problems similar to the one we just solved:

1. Solve: 5x + 3 = 3x + 1 = 7
2. Solve: 10x - 2 = 4x + 6 = 12
3. Solve: 6x + 1 = 2x - 3 = 5

Remember to break each problem into two separate equations and solve for 'x' in each. Compare your solutions to see if there is a consistent value for 'x' that satisfies the entire original equation. Good luck, and happy solving!