Multiply Integers & Solve Equations: Math Problems
Hey guys! Let's dive into some integer multiplication and equation-solving fun. We've got some cool problems lined up, so grab your thinking caps and let's get started!
Multiplying Integers
First off, we're tackling multiplication with positive and negative numbers. Remember the basic rules: a positive times a positive is positive, a negative times a negative is also positive, and a positive times a negative (or vice versa) is negative. Keeping these rules in mind will make these problems a breeze.
Problem E: 4 x (-3) x 5
Okay, let's break down this problem: 4 x (-3) x 5. The key here is to tackle it step by step.
- First, multiply 4 by -3. Since we have a positive number times a negative number, the result will be negative. 4 times 3 is 12, so 4 x (-3) = -12.
- Now, we have -12 x 5. Again, we’re multiplying a negative number by a positive number, so the result will be negative. 12 times 5 is 60, so -12 x 5 = -60.
So, the final answer for 4 x (-3) x 5 is -60. See? Not so tough when you break it down. Remember, always pay attention to the signs!
When dealing with integer multiplication, especially with multiple factors, it’s super important to keep track of the signs. Start by multiplying the first two numbers and then multiply the result by the next number, and so on. This step-by-step approach helps avoid confusion and ensures you get the correct answer. For example, in this problem, multiplying 4 and -3 first gives us -12. Then, multiplying -12 by 5 gives us -60. Breaking it down like this makes the process much more manageable and less prone to errors. It's also a great habit to get into, as it will help you tackle more complex problems later on. Think of it like building blocks – each step is a block that builds towards the final answer.
Problem F: -5 x (-7) x (-2)
Next up, we have -5 x (-7) x (-2). This one has three negative numbers, so let’s see how that works out.
- First, let’s multiply -5 by -7. Remember, a negative times a negative is a positive. 5 times 7 is 35, so -5 x (-7) = 35.
- Now, we have 35 x (-2). Here, we’re multiplying a positive number by a negative number, so the result will be negative. 35 times 2 is 70, so 35 x (-2) = -70.
Therefore, the final answer for -5 x (-7) x (-2) is -70. Notice how the two negatives canceled each other out in the first step, but the third negative made the final result negative. Keep this pattern in mind!
When you have a series of integers to multiply, an easy way to figure out the sign of the final answer is to count the number of negative integers. If there's an even number of negative integers, the final answer will be positive because each pair of negative numbers multiplies to a positive. If there's an odd number of negative integers, the final answer will be negative. In this case, we had three negative integers, which is an odd number, so we knew the final result would be negative. This trick can save you time and help you double-check your work. It’s like a quick mental shortcut that helps you navigate through the problem more efficiently.
Solving Equations: Finding the Missing Number
Now, let's switch gears to solving equations. These problems ask us to find the missing number that makes the equation true. The key here is to isolate the variable (the missing number) by performing the inverse operation. For example, if we have multiplication, we use division to undo it. Let’s jump into it!
Problem A: -5 x ... = 35
We need to find a number that, when multiplied by -5, equals 35. Think: what number times 5 gives us 35? It’s 7, right? But we need the result to be positive 35, and we're multiplying by a negative number. So, the missing number must be negative.
-5 x (-7) = 35
So, the missing number is -7. Simple, right? You just have to think about the signs and the multiplication facts you already know.
When solving equations like this, it's helpful to think of multiplication and division as opposite operations. If you're trying to find a missing factor in a multiplication problem, you can divide the result by the known factor. For instance, in this case, you can divide 35 by -5 to find the missing number. Remember that dividing a positive number by a negative number results in a negative number, and vice versa. This inverse relationship between multiplication and division is a fundamental concept in algebra and makes solving equations much easier. By understanding this, you can approach these problems with confidence and accuracy.
Problem B: 9 x ... = -72
Here, we need to find a number that, when multiplied by 9, equals -72. We know 9 times 8 is 72, but we need a negative 72. Since we're multiplying a positive number (9) by the missing number, the missing number must be negative to get a negative result.
9 x (-8) = -72
So, the missing number is -8. Keep practicing, and these will become second nature!
Understanding the relationship between factors and products is essential when solving these types of equations. The factors are the numbers you multiply together, and the product is the result. In this problem, 9 is one factor, the missing number is the other factor, and -72 is the product. To find the missing factor, you need to think about what number, when multiplied by 9, gives you -72. This involves recalling your multiplication facts and considering the sign rules. If the product is negative and one factor is positive, the other factor must be negative. This logical thinking is key to solving these equations quickly and efficiently.
Problem C: 5 - 6 x ... = 23
This one's a bit trickier because we have subtraction and multiplication. Remember the order of operations (PEMDAS/BODMAS): we do multiplication before subtraction. So, we need to isolate the term with the missing number first.
Let's rewrite the equation: 5 - 6 x ... = 23
First, we want to isolate the -6 x ... part. To do that, we subtract 5 from both sides of the equation:
5 - 6 x ... - 5 = 23 - 5
-6 x ... = 18
Now, we need to find a number that, when multiplied by -6, equals 18. We know 6 times 3 is 18, but we need a positive 18, and we're multiplying by a negative number. So, the missing number must be negative.
-6 x (-3) = 18
Wait a minute! There's a mistake here. -6 multiplied by -3 will equal +18. We are looking for -6 multiplied by something to give us 18. So it must be a negative number that turns it into a positive. So, -6 x -3 equals 18. The correct way to solve this would be to divide 18 by -6, which is -3. Let's plug this back into the equation: 5 - 6 * (-3) = 5 - (-18) which equals 5 + 18 which gives us 23. Bingo!
So, the missing number is -3. See how important it is to check your work?
When dealing with more complex equations like this one, it's crucial to follow the order of operations and break the problem down into smaller, manageable steps. The order of operations (PEMDAS/BODMAS) dictates that you perform operations in the following order: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). In this case, we needed to address the multiplication before we could deal with the subtraction. Additionally, remember that performing the same operation on both sides of the equation maintains the equality. This is a fundamental principle in algebra and allows you to isolate the variable and solve for the missing number. Taking each step methodically and double-checking your work helps prevent errors and ensures you arrive at the correct solution.
Problem D: 13 x ... = -39
We need to find a number that, when multiplied by 13, equals -39. We know 13 times 3 is 39, but we need a negative 39. Since we're multiplying a positive number (13) by the missing number, the missing number must be negative to get a negative result.
13 x (-3) = -39
So, the missing number is -3. Feeling more confident now?
Recall your multiplication facts and understanding of the sign rules for multiplication. When you know that 13 times 3 equals 39, you're already halfway to the answer. The key then is to determine the correct sign. Because the product (-39) is negative and one factor (13) is positive, the other factor must be negative. Therefore, the missing number is -3. This problem highlights how recalling basic multiplication facts combined with sign rules can quickly lead to the solution. Practice these mental steps, and you’ll find you can solve these problems with increasing speed and accuracy.
Problem E: ... x 3 = -12
This time, the missing number is at the beginning, but the process is the same. We need to find a number that, when multiplied by 3, equals -12. We know 4 times 3 is 12, and we need a negative 12. Since we're multiplying by a positive 3, the missing number must be negative.
(-4) x 3 = -12
So, the missing number is -4. You’re getting the hang of this!
Understanding the commutative property of multiplication can be very helpful here. The commutative property states that the order in which you multiply numbers does not change the product (a x b = b x a). In this problem, whether the missing number is written before or after the 3 doesn’t change how you solve it. You still need to find a number that, when multiplied by 3, gives you -12. This understanding can simplify problems where the unknown is in a less familiar position. It’s all about recognizing the underlying mathematical relationship and applying the rules of arithmetic consistently.
Problem F: -174 x ... = -2
Okay, this one looks a little intimidating with the big number, but don't worry! The principle is the same. -174 times what gives you -2? Let’s think...
If we divide -2 by -174, we can get the missing number.
-2 / -174 = 1/87
So, the missing number is 1/87. It might be a fraction, but we solved it!
When dealing with problems involving larger numbers, it's often helpful to consider whether the missing number might be a fraction or a decimal. In this case, realizing that the product (-2) is much smaller in magnitude than one of the factors (-174) suggests that the missing number must be a fraction. The process of dividing the product by the known factor remains the same, but the answer will be expressed as a fraction. Problems like these reinforce the understanding that solutions to equations aren't always whole numbers and that working with fractions is a necessary skill in algebra.
Problem G: -7 x ... = 63
We need to find a number that, when multiplied by -7, equals 63. We know 7 times 9 is 63, but we need a positive 63. Since we're multiplying by a negative -7, the missing number must be negative.
-7 x (-9) = 63
So, the missing number is -9. We’re almost through all the problems!
Again, leveraging your multiplication facts is key. Knowing that 7 times 9 equals 63 immediately gets you close to the solution. Then, it’s about applying the sign rules correctly. In this case, a negative times a negative results in a positive, so -7 multiplied by -9 gives you 63. This problem reinforces the importance of both numerical facts and sign conventions in solving equations. With each problem you solve, you’re building your confidence and accuracy in applying these concepts.
Problem H: 4 x ... = -10
Last one! We need to find a number that, when multiplied by 4, equals -10. We know 4 times 2 is 8 and 4 times 3 is 12, so the answer may not be a simple integer, so let's try dividing -10 by 4.
-10/4 = -5/2 or -2.5
So, the missing number is -2.5. We nailed it!
This problem demonstrates that not all solutions are integers, and it's perfectly acceptable to have fractional or decimal answers. When the product (-10) is not a direct multiple of the known factor (4), you need to consider that the missing number might be a non-integer. Dividing -10 by 4 gives you -2.5, which is the missing number. Problems like these broaden your understanding of number types and reinforce the importance of being comfortable working with fractions and decimals. They also show that mathematical solutions can come in various forms, and it's essential to be adaptable in your approach.
Conclusion
And there you have it! We’ve solved some integer multiplication problems and found the missing numbers in several equations. Remember the rules for multiplying positive and negative numbers, and always double-check your work, especially when dealing with negatives. Keep practicing, and you’ll become a math whiz in no time. Keep rocking, mathletes!