Tutor profile: Hadley B.
Subject: Basic Math
If Amy has 3 eggs and Hector has 2 eggs, how many eggs to Amy and Hector have combined?
If we lay out Amy's eggs (we will be using stars to represent eggs) she has 3: * * * If we lay out Hector's eggs, he has 2: * * If we combine Amy and Hector's eggs we have this: * * * * * If we count those combined eggs, we have 5 total eggs. So Amy and Hector have 5 eggs combined.
Find f'(x) for f(x) = 4x^5 + 9 using the Power Rule.
The Power Rule is defined as (d/dx)(x^n) = n*x^(n - 1). Looking at our f(x), 9 is a constant so we know the derivative of that is 0 and we are left with 4x^5 to use the Power Rule on. In this case, our n is 5, so we bring that in front of the x term and multiply it. x's coefficient will now be 5*4 = 20. According to the Power Rule, we also have to subtract one from the exponent so the exponent becomes 5-1 = 4. This means that our f'(x) = 20x^4 + 0. This could also be written as f'(x) = 20x^4.
The sum of 30 times x and 15 times y equals 105. The sum of 4 times x and 10 times y equals 38. What are the values of x and y?
This is a system of equations that can be written as: 30x + 15y = 105 and 4x + 10y = 38. The first step to solving a system of equations is to find one variable in terms of the other. Using the second equation, we can solve for y in terms of x. By subtracting 4x from both sides, we will have 10y = 38 - 4x. Next, to get y by itself, we divide both sides by 10 to get y = (38 - 4x)/10. The next step is to plug this value of y in the first equation so we will only have x's. That equation becomes: 30x + 15*((38 - 4x)/10) = 105. To get rid of the fraction, we can multiply everything by 10 to get: 300x + 15*(38 - 4x) = 1050. The next thing we need to do is combine the like terms. Before we do that we need to distribute the 15 to 38 - 4x. To do this we multiply each term by 15: (15*38) - (15*4x) = 570 - 60x. Now our equation looks like this: 300x + 570 - 60x = 1050. To combine like terms, we do (300x - 60x) and subtract 570 from each side so the constants are combined. Our equation now looks like this: 240x = 480. Now we divide both sides by 240 to get x by itself and now we know x=2. Lastly, we plug x into the equation that gave y in terms of x to solve for y. We can recall that y = (38 - 4x)/10. and when we plug 2 in for x: y = (38 - (4*2))/10. Simplify the numerator: 38 - 8 = 30. So now we have y = 30/10 which means y = 3. So now we have solved this system of equations and found that x = 2 and y = 3! To check your work, you can plug these values into the original, given equations and make sure they are true. (30*2) + (15*3) = 60 + 45 = 105. True! (4*2) + (10*3) = 8 + 30 = 38. True!