False Solutions: Solving 9m - 14 = 4m + 56

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Hey math whizzes! Today, we're diving into a fun little problem that's all about testing our equation-solving skills and our ability to spot those tricky false solutions. We've got a linear equation: 9m - 14 = 4m + 56, and a handy-dandy solution set S: {5, 14, 70, 112}. Our mission, should we choose to accept it (and we totally should!), is to figure out which of these numbers, when plugged into our equation, don't make it true. That's right, we're looking for the imposters, the ones that break the equality! We'll also show you all the work, step-by-step, so you can follow along and really nail this concept. So, grab your notebooks, maybe a snack, and let's get this math party started!

The Equation and Our Suspects

Alright guys, let's break down what we're dealing with. Our main equation is 9m - 14 = 4m + 56. This is a linear equation, meaning the highest power of our variable 'm' is 1. Our goal when solving equations like this is to isolate the variable 'm' on one side of the equals sign. We do this by using inverse operations – whatever we do to one side, we must do to the other to keep the equation balanced. Think of it like a perfectly balanced scale; you can't just remove weight from one side without affecting the whole thing!

Now, let's look at our potential solution set, S: {5, 14, 70, 112}. These are the candidates we need to test. The problem asks us to find which of these numbers, when substituted for 'm', will make the equation false. This means we're specifically looking for the numbers that don't satisfy the equality. It's a bit like a detective case where most of the clues point to one suspect, but we need to find the one who didn't do it, or in this case, the number that doesn't solve the equation. We're not just solving for 'm' and picking the answer; we're checking each option against the original equation. This is a super important skill because in real-world scenarios, you might be given a set of possible answers, and you need to verify them.

Solving the Equation First (Our True North)

Before we start plugging in numbers, it's always a good idea to find the actual solution to the equation. This will give us a benchmark. If one of our suspect numbers matches the actual solution, we know it should make the equation true. The others, if they don't match, are our potential false solutions. So, let's solve 9m - 14 = 4m + 56.

Step 1: Get all the 'm' terms on one side. I like to move the smaller 'm' term to avoid dealing with negative coefficients if possible. So, let's subtract 4m from both sides:

9m - 4m - 14 = 4m - 4m + 56
5m - 14 = 56

See? We've combined the 'm' terms. Now we have 5m - 14 = 56.

Step 2: Get the constant terms on the other side. Our goal is to get the term with 'm' by itself. To undo the '- 14', we'll add 14 to both sides:

5m - 14 + 14 = 56 + 14
5m = 70

Looking good! We're almost there. We now have 5m = 70.

Step 3: Isolate 'm'. The 'm' is currently being multiplied by 5. To isolate 'm', we'll do the inverse operation: divide both sides by 5:

5m / 5 = 70 / 5
m = 14

Boom! We found the actual solution to the equation. The value of 'm' that makes 9m - 14 = 4m + 56 true is 14.

Now that we know the true solution is 14, we can look at our solution set S: {5, 14, 70, 112}. We already know that 14 should make the equation true because it's the actual solution. The question asks for the integers that make the equation false. This means we need to check the other numbers in the set: 5, 70, and 112. We expect these three numbers not to satisfy the equation.

Testing the Suspects: Finding the False Solutions

Alright, team, it's testing time! We're going to take each number from our suspect list (excluding the one we know is correct, 14) and plug it into the original equation 9m - 14 = 4m + 56. Remember, if the left side does NOT equal the right side, then that number is one of our false solutions.

Testing m = 5

Let's substitute m = 5 into the equation:

  • Left Side (LS): 9m - 14 = 9(5) - 14 = 45 - 14 = 31
  • Right Side (RS): 4m + 56 = 4(5) + 56 = 20 + 56 = 76

Does LS = RS? 31 ≠ 76. So, m = 5 does not make the equation true. It makes it false! This means 5 is one of our answers. High five!

Testing m = 14

We already solved the equation and found that m = 14 is the correct solution. But let's double-check just to be absolutely sure and to demonstrate how the true solution works.

  • Left Side (LS): 9m - 14 = 9(14) - 14 = 126 - 14 = 112
  • Right Side (RS): 4m + 56 = 4(14) + 56 = 56 + 56 = 112

Does LS = RS? 112 = 112. Yes! As expected, m = 14 makes the equation true. This is not one of the numbers we're looking for.

Testing m = 70

Now let's try m = 70.

  • Left Side (LS): 9m - 14 = 9(70) - 14 = 630 - 14 = 616
  • Right Side (RS): 4m + 56 = 4(70) + 56 = 280 + 56 = 336

Does LS = RS? 616 ≠ 336. Nope! So, m = 70 does not satisfy the equation. This is another false solution. We're on a roll!

Testing m = 112

Finally, let's test m = 112.

  • Left Side (LS): 9m - 14 = 9(112) - 14 = 1008 - 14 = 994
  • Right Side (RS): 4m + 56 = 4(112) + 56 = 448 + 56 = 504

Does LS = RS? 994 ≠ 504. And just like that, m = 112 also fails to make the equation true. This is our third false solution!

Conclusion: The False Prophets

So, guys, after all that hard work and plugging in numbers, we've identified which integers from the set S: {5, 14, 70, 112} make the equation 9m - 14 = 4m + 56 false. We found the true solution to be m = 14. When we tested the other numbers:

  • m = 5: Left side was 31, right side was 76. False.
  • m = 70: Left side was 616, right side was 336. False.
  • m = 112: Left side was 994, right side was 504. False.

Therefore, the integers in the solution set S that make the equation false are 5, 70, and 112. It's crucial to remember that solving an equation means finding the value(s) that make it true. In this problem, we were specifically asked to find the ones that don't make it true. This exercise highlights the importance of both solving an equation correctly and verifying potential solutions, especially when dealing with multiple-choice scenarios or when checking your work. Keep practicing, and you'll become a master equation detective in no time!