Simplify Expressions & Solve Math Problems | 6th Grade
Hey guys! Let's break down these math problems step by step. We've got a mix of simplifying expressions and some basic arithmetic. So, grab your pencils, and let's dive in!
Simplifying Expressions
1) -30a - (-56)
Okay, so when you see a minus sign in front of a parenthesis with a minus sign inside, it's like saying "minus a negative," which turns into a positive. So, -(-56) becomes +56. Therefore, the expression simplifies to:
-30a + 56
That's it! We can't simplify it any further because -30a and 56 are not like terms (one has a variable a, and the other is just a number).
When dealing with algebraic expressions, remember that combining like terms is the key. You can only add or subtract terms that have the same variable raised to the same power. In this case, -30a is a term with the variable a, while 56 is a constant term. Since they are different types of terms, we cannot combine them any further. This principle is fundamental in algebra and helps in simplifying complex equations and expressions. Understanding like terms is the cornerstone for manipulating and solving algebraic problems effectively. For instance, if we had another term like -10a, we could combine it with -30a to get -40a. However, since we don't have any other a terms, the expression -30a + 56 remains as it is. Therefore, always look for like terms when simplifying expressions, ensuring that you only combine terms that have the same variable and exponent.
2) -33a + 6.5b
In this expression, we have two terms: -33a and 6.5b. Notice that one term has the variable a and the other has the variable b. Since a and b are different variables, these terms are not like terms. Therefore, we cannot combine them. The expression is already in its simplest form:
-33a + 6.5b
Understanding the concept of unlike terms is crucial when simplifying algebraic expressions. Unlike terms are terms that have different variables or the same variables raised to different powers. As such, they cannot be combined through addition or subtraction. Recognizing unlike terms ensures that you don't make the mistake of combining terms that shouldn't be combined, which would lead to an incorrect simplification. For instance, in the expression -33a + 6.5b, -33a is a term with the variable a, while 6.5b is a term with the variable b. Because these variables are different, these terms are unlike and cannot be combined. This concept is not just limited to single variables; it also applies to terms with exponents. For example, x^2 and x^3 are unlike terms because the variable x is raised to different powers. Thus, mastering the ability to identify and differentiate between like and unlike terms is essential for successful algebraic manipulation and simplification.
3) 2a + 14 - (5)
Here, we can simplify the numerical part of the expression. We have 14 - 5, which equals 9. So, the expression becomes:
2a + 9
Again, we cannot simplify it further because 2a and 9 are not like terms.
When simplifying expressions, the order of operations is extremely important. In the expression 2a + 14 - (5), we first deal with the parenthesis. In this case, it’s a simple subtraction within the expression: 14 - 5. Calculating this gives us 9. So, the expression simplifies to 2a + 9. Now, we observe that 2a and 9 are not like terms. The term 2a contains the variable a, whereas 9 is a constant term without any variables. Since they are not like terms, they cannot be combined. This means that the expression 2a + 9 is already in its simplest form. Remember that simplifying expressions often involves reducing the number of terms to their minimum by combining like terms, which is achieved by paying attention to the variables and constants in the expression. The expression 2a + 9 is a clear and concise way to represent the original expression after simplification.
4) 0.56 * 1.3 * (-11a)
First, let's multiply the numbers: 0.56 * 1.3 = 0.728. Then, we multiply that result by -11: 0.728 * -11 = -8.008. So, the expression simplifies to:
-8.008a
When dealing with numerical coefficients in algebraic expressions, it is essential to perform the arithmetic operations correctly. In the given expression 0.56 * 1.3 * (-11a), the numerical coefficients are 0.56, 1.3, and -11. The first step is to multiply 0.56 by 1.3, which yields 0.728. Next, we multiply 0.728 by -11, which results in -8.008. This multiplication gives us the simplified numerical coefficient for the variable a. So, the simplified expression is -8.008a. Always ensure you follow the correct order of operations and pay attention to the signs of the numbers. A negative number multiplied by a positive number results in a negative number. Precision in these calculations is crucial to arrive at the correct simplified expression. By accurately performing these multiplications, we ensure the algebraic expression is correctly represented with its simplest numerical coefficient.
5) -26 - (-3) - 125
Remember, subtracting a negative number is the same as adding its positive counterpart. So, -(-3) becomes +3. The expression is now:
-26 + 3 - 125
Now, let's add and subtract from left to right: -26 + 3 = -23. Then, -23 - 125 = -148. So,
-148
When performing arithmetic operations with negative numbers, it’s crucial to understand the rules of addition and subtraction. The expression -26 - (-3) - 125 involves both subtraction of a negative number and subtraction of a positive number. First, we deal with -(-3), which is equivalent to adding 3. So, the expression becomes -26 + 3 - 125. Next, we perform the addition: -26 + 3 = -23. Finally, we subtract 125 from -23, which gives us -23 - 125 = -148. Understanding these rules is essential for accurately simplifying the expression. A common mistake is to mishandle the negative signs, which can lead to incorrect results. Always remember that subtracting a negative number is the same as adding the positive number, and subtracting a positive number is straightforward subtraction. This meticulous attention to detail ensures the correct simplification of arithmetic expressions involving negative numbers.
6) 5 - (-756)
Similar to problem 5, subtracting a negative is the same as adding a positive:
5 + 756 = 761
So,
761
Understanding the concept of subtracting a negative number is vital in basic arithmetic. The expression 5 - (-756) involves subtracting a negative number, which transforms into adding its positive counterpart. Therefore, the expression becomes 5 + 756. Performing this addition is straightforward: 5 + 756 = 761. This principle is derived from the fundamental rules of arithmetic, where subtracting a negative is equivalent to moving in the positive direction on the number line. The expression simplifies to a single positive number, 761. A solid understanding of this concept prevents common errors and ensures that arithmetic operations involving negative numbers are executed accurately. This foundational knowledge is crucial for progressing to more complex mathematical problems.
7) -88ab - 9.3
Here, we have two terms: -88ab and -9.3. These terms are not like terms because one contains the variables ab and the other is a constant. Therefore, we cannot simplify this expression any further.
-88ab - 9.3
In identifying like and unlike terms in algebraic expressions, it's essential to examine the variables and their exponents. The expression -88ab - 9.3 consists of two terms: -88ab, which contains the variables a and b, and -9.3, which is a constant term. Since -88ab includes variables and -9.3 does not, they are considered unlike terms. Unlike terms cannot be combined through addition or subtraction. Therefore, the expression -88ab - 9.3 is already in its simplest form. Understanding this distinction is crucial to avoid incorrectly combining terms that should not be combined. Algebraic expressions are simplified only when like terms are combined. If there are no like terms, the expression remains as is. This principle is fundamental in algebra and helps to accurately simplify more complex mathematical expressions.
8) 3.1 - (-6.6) * (-4ab)
First, let's deal with the multiplication part. We have -6.6 * -4ab. Multiplying -6.6 by -4 gives us 26.4. So, the term becomes 26.4ab. Now, our expression is:
3.1 - 26.4ab
These terms are not like terms (one is a constant, and the other has variables), so we cannot simplify further.
3. 1 - 26.4ab
When dealing with combined operations in algebraic expressions, it's crucial to follow the correct order of operations (PEMDAS/BODMAS). In the expression 3.1 - (-6.6) * (-4ab), we first address the multiplication part: (-6.6) * (-4ab). Multiplying -6.6 by -4 results in 26.4. Therefore, the term becomes 26.4ab. Now, the expression is 3.1 - 26.4ab. Next, we examine the terms 3.1 and -26.4ab. These are unlike terms because 3.1 is a constant term without any variables, while -26.4ab includes the variables a and b. Since they are unlike terms, they cannot be combined. As a result, the expression 3.1 - 26.4ab remains as is, and we cannot simplify it any further. Always remember to perform multiplication before addition or subtraction and to combine only like terms to accurately simplify algebraic expressions.
Problem 539 (6th Grade)
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