Tutor profile: Cassie Q.
Solve the following problem: (2+ 7*4) - 6 + 3
The problems done out step-by-step would look like this: 1. (2+ 7*4) -6 + 3 2. (2 + 28) -6 + 3 3. 30 - 6 + 3 4. 24 + 3 5. 27 (our problem equals 27) This question concerns ORDER OF OPERATIONS. When solving or simplifying problems that include more than one operation, we must solve them in the correct order to get the right answer; if we do the operations out of order, we will end up with the wrong answer. The order of operations goes like so: 1) Parenthesis 2) Exponents 3) Multiplication 4) Division 5) Addition 6) Subtraction PEMDAS is a commonly used acronym to represent the order of operations. It is important to remember, however, that multiplication & division, and addition & subtraction are INTERCHANGEABLE. That means after you do your parenthesis and exponents, if your equation includes BOTH multiplication and division, you will simply do whichever comes first (it does not have to be multiplication just because it comes first in PEMDAS). The same is true for addition and subtraction; when both are used in an equation and you have gone through your parenthesis, exponents, multiplication and division, you simply do whichever comes first in the problem (you do not have to do addition first just because PEMDAS has it first. Our problem reads: (2 + 7*4) - 6 + 3. First, we see if we have any parenthesis in our problem. And yes, we have parenthesis around our (2 + 7*4). This means we have to do the operations INSIDE the parenthesis before anything else. Within the parenthesis we see that we have two operations happening: addition and multiplication. So which do we do first? Well, according to PEMDAS, we multiply before we add. Therefore, we first do 7*4, which equals 28. The inside of our parenthesis will now look like (2 +28). With our multiplication done, we can now do the addition to get 30. The inside of our parenthesis is now just (30) which means we can drop the parenthesis because we have completed all of the operations inside of them. Our problem now reads: 30 - 6 + 3. Next, we see if our problem has any exponents. It does not so next we check for multiplication and division. There is no more multiplication to be done, and no division necessary, so we can keep moving through PEMDAS. Finally, we check for addition and subtraction. REMEMBER, when dealing with addition and subtraction, the order does not matter; we simply do what comes first in our equation. So even though addition technically comes first in PEMDAS, subtraction comes first in our problem and is therefore done first. So our next step is 30 - 6 which equals 24. Our problem now reads 24 + 3. Here we simply add the two together and get a final answer of 27.
Who is considered the father of Functionalism?
Put the following equation into slope-intercept form: -6y + 3x = 18
The first step in solving this problem is to write down what slope-intercept form looks like: y = mx + b Just done out, the steps to the problem would look like this: 1) -6y + 3x = 18 - 3x = -3x 2) -6y = 18 - 3x / -6 = /-6; /-6 (/-6 = divide by -6) 3) y = -3 + (3/6)x 4) y = -3 + (1/2)x 5) y = (1/2)x - 3 (COMPLETE--we are in slope-intercept form) A detailed explanation: Our goal, therefore, is to get 'y' alone by somehow getting rid of the '3x' and the '-6'. To do this, we will use opposite operations to move each of these to the other side of the equal sign. The most important thing to remember in these types of problems, is that what you do to one side of the equation, you MUST also do to the other side of the equation. So whatever we do to the "-6y +3x" side (whether it is addition/subtraction or multiplication/division), we will do the EXACT same thing to the "18" side. Even though we could start with either the '3x' or the '-6', the easiest option is to start with '3x'. Why? To understand why look at what operations each of these entail: the '-6' serves as a coefficient to the 'y', meaning it is a constant number being MULTIPLIED with a variable (y). The opposite of multiplication is division so we would have to divide EACH side by '-6' to eliminate the '-6' from the left side of the equation. The '3x' on the other hand is being ADDED to the '-6y', and the opposite of addition is SUBTRACTION. Here, it will be much easier to start with subtraction than to start with division. As stated earlier, to move a number from one side of the equation to the other, we must used OPPOSITE operations on EACH side of the equation (what we do to one side we must always do to the other). So starting with the easier of the two, to eliminate '3x' from the left side, we must SUBTRACT '3x' from each side. Remember 3 is a coefficient to 'x', meaning it is being MULTIPLIED, so it can only be separated from the 'x' using DIVISION. BUT, we are using SUBTRACTION, therefore the 3 MUST stay with the 'x'. If we only subtract '3' or 'x' from each side, we will come out with an incorrect answer. Keeping this in mind, when we subtract '3x' from the left side of the equation, we end up doing "3x - 3x", which equals 0, thus eliminating it from the left side. Now we move on to the right side of the equation because whatever we do to one side of the equation, we MUST do to the other. So on the right side we end up with "18 - 3x'. It may be tempting to say, "18 -3 = 15, so this becomes 15x", but this is INCORRECT. 18 and '3x' are NOT like terms and therefore CANNOT be combined-they must be kept separate. Picture like terms as relationships. When you have just a number, it is like someone who is single. When you have a number that is acting as the coefficient to a variable (aka any number directly followed by a letter), it is like someone who is married. Their relationship statuses are not alike. In terms of the problem, say 18 (our single person) sees the '3' (one half of our married couple) and wants to date them. BUT, 18 cannot date 3 because 3 they are not alike; 18 may be single but 3 is part of a couple with 'x' and those two cannot be broken up. In our problem, 18 may want to be combined with 3 BUT it cannot because 3 they are not like terms; 18 is a constant number BUT 3 is already being combined to 'x' and cannot be combine with 18. So while it may be tempting to subtract '3x' from the right side and think the result is '15x', we cannot do so. Instead, the right side of the equation simply becomes "18 - 3x'. At this point in the problem, our equation should look like this: -6y = 18 - 3x. Next, we will move the -6 from the left side to the right side. As stated earlier, -6 is being multiplied with 'y' so it must be DIVIDED FROM EACH SIDE. It is necessary to divide by '-6' instead of just '6' because we need 'y' to be POSITIVE to get the correct slope-intercept formula. As 'y' is now negative, we must divide it by another negative to make it positive. Our left side of the equation would then go like this: "-6y / -6" --> -6 / -6 = '1y' which we would just write 'y' for. Note that we DO NOT divide by '-6y', we only divide by '-6'--this is because we WANT the 'y' to stay on that left side to form our slope-intercept formula. With our left side of the formula complete, we now repeat our operation on the right side of the equation. On the other side, we divide 18 by -6 AND '3x' by -6. First doing out "18 / -6" we get "-3" --> because 18 is positive and -6 is a negative, a positive divided by a negative gives us a negative number. Next, we do our "-3x / -6". Because we are working with DIVISION, we do not have to worry about like terms here. While we could leave this as it is written, slope-intercept form should always use the SIMPLEST numbers possible. So looking at "-3x / -6", what can be simplified? Well, we know that a negative divided by a negative is just a positive, so we can eliminate our negative signs, giving us "3x / 6". Now, we ask ourselves if the fraction can be simplified in any way. Is there any number that we can divide BOTH 3 AND 6 by, to give us smaller whole numbers (ONLY whole numbers should be used in simplified fractions)? The answer is yes: we can divide each of these by 3 . 3/3 = 1, and 6/3 = 2. Therefore, our fraction can be simplified to "(1/2)x". Our equation now looks like this: y = -3 + (1/2)x To get it into our y = mx +b form, we simply change the order of the right side of the equation to give us: y = (1/2)x - 3 --> We are now in slope-intercept form and our solution is complete.
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