Mike S. answered • 10/09/15
Tutor
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Math Tutoring from 1 to Calculus
1) Add the equations together. This eliminates the "y". We have to reduce the number of variables to one in order to solve the equations.
This yields 3x = 18, therefore X = 6. If X = 6, then y = -7. We know this by substituting 6 back into either of the original equations for X and solving for y.
2) Let b = Bobby's age and a = Arianna's age
b = 2a + 3 (3 years older than twice "a")
b + a = 9 (Total of their ages is 9)
(2a + 3) + a = 9
...Substituted 2a + 3 for b in the second equation since they are equal.
See if you can get it from here.
3) Substitute x^2 + 5 for y into the second equation to get...
6x - (x^2 + 5) = 3 (We did this to reduce to only one variable.)
This is a quadratic equation since it has a squared term, therefore you need to factor it, complete the square, or use the quadratic formula to solve it. (Whichever is easier, but I try to avoid the quadratic formula.)
Distributing the "-" through the parentheses and writing the terms in descending order helps...
-x^2 + 6x - 5 = 3
I would also rearrange this to make the x^2 term positive and set the equation equal to 0. In order to factor and solve this, it should be equal to 0.
x^2 - 6x + 8 = 0 (Make sure you understand how we got here. You can do anything you want to an equation as long as you do the same thing to both sides to keep it balanced.)
Having fun yet? ....me neither, but it is better than cleaning a shower drain is it not? ... Stay positive.
Alright, now factor x^2 - 6x + 8 = 0 into two binomials. FOIL got us into this mess. Reverse the process to get out. Ask yourself, "What numbers multiply to get 8 and add to get -6?" Try, try again.
(X - 2)(X - 4) = 0
Since any number times 0 = 0, then this equation is true if
(X - 2) = 0 Or (X - 4) = 0
So the answers are X = 2 or X = 4. Substitute both of these back into the original equations to find their corresponding "y" values. Check to see if these answers work in both of the original equations. If they do not, then cry.
If X = 2, then y = 9. If X = 4, then y = 21. That can also be written as (2,9) or (4,21).
Yes, they satisfy both original equations. Whew.
You're right. The shower drain looks more attractive now. Nobody likes a heavy dose of vegetables either but they are also good for your brain in ways that are hard to explain.
Iulia K.
Whatever Anne's age may be, Bobby is twice her age, plus three more years.
You can represent this with B = 2A + 3
Next, the problem tells us that the ages of Bobby and Anne add up to 9, so if you add up Bobby and Anne's ages, you get 9.
(2A + 3) + A = 9
You want to show Bobby's age as 2A + 3 because to solve an equation by its self you can only have one variable. (If you want to solve two equations then you need two variables, if you want to solve three equations, then you need three variables).
(2A + 3) + A = 9 Now add up the As.
3A + 3 = 9
3A + 3 = 9 Now subtract 3 from both sides.
-3 -3
3A + 0 = 6
3A = 6
3A = 6 Now divide both sides by 3.
3 3
A = 2 Now we know that Arianna is 2 years old.
We also know that Bobby is three year's older than twice Arianna's age, so twice Arianna's age is 4, and three years older than that is 7.
Check your answer: does 2+7 = 9? Yes! Yes it does.
3. For this problem you have two kinds of equations. One that is a parabola, so it's kind of shaped like a U or an upside down U), and linear equation that will just be a line that will go diagonally across the graph.
(The quadratic equation is the one with the X raised to the power of 2, and the linear equation is the one with X not raised to any special power.)
Luckily, the quadratic equation already has the y on one side so you don't have to worry about it. For the linear equation though, we need to get the y be itself on one side first.
6x - y = 3 Start with 6x - y = 3 and move the 6x to the other side of the equation by subtracting 6x from both sides.
6x - y = 3
-6x -6x
0x - y = -6x + 3
-y = -6x + 3 Now multiply both sides of the equation by negative 1 to make the y positive.
-1(-y) = (-6x + 3)(-1)
y = 6x -3 ← This is your answer.
Now set the problems equal to each to solve for the Xs, which will tell you where the two graphs (the parabola and the straight line) will intersect.
x2 + 5 = 6x -3 Because this is a quadratic equation you need to put everything on one side of the equation to solve for it. It looks like this.
x2 + 5 - 6x + 3 = 0 Here I subtracted the 6x from both sides to get a negative 6x on the left, and I added 3 to both sides, to get a positive three also on the left. Notice that the equation is equal to 0.
Before you solve for the equation, you need to combine the like terms (so combine the x2 with the x2, the Xs with the Xs, and the plain number with the plain numbers).
Like terms are similar to each other because they have all of the same variables raised to the same powers so
x4y3 and -5x4y3 are like terms because they are raised to the same powers. It doesn't matter what the numbers they are multiplied by are, but what allows you to combine like terms is that the variables are raised to the same power/powers.
If you want to combine x4y3 and -5x4y3 for example, you get -4x4y3.
There is only 1 term for x2 though, so you can't combine that with any other terms of x2, there is only term for x, so you can't combine that with any other terms of x, but there are two plain terms, 5 and 3 so you combine them together to make 8.
In other words: x2 - 6x + 8 = 0
Now you have two options, solve for the equation by factoring, or by the quadratic formula. Let's solve by factoring.
Break up the equation into its two factors:
x2 - 6x + 10 = 0 → (x - 4) (x - 2) = 0 This is because when you multiply the Xs together you get 4, when you multiply just the two numbers together (-4 and -2) you get 8.
↓
When you multiply the first x by -2, you get -2x and when you multiply the second x by -4 you get -4x, and when you add -4x + -2x, you get -6x like in the equation x2 - 6x + 8.
Now because this is a quadratic equation (where you have x2) there will be two answers. (That means that on the graph, the straight line will pass through the U shape at two different points, diagonally in this case.)
This is because you have to Xs and each one will give you a different answer.
To find the value of the first x, first divide both sides of the equation by (x - 2).
(x - 4) (x - 2) = 0 Notice 0 divided by any number is 0 so 0 is 0.
(x -2) (x-2) (x-2)
This let's us solve for (x -4) in peace now because it's equal to 0.
x - 4 = 0 and to solve for x you need to add 4 to both sides of the equation.
+4 +4
Thus x = 4 is one of your answers.
Now do the same thing to solve for the x in the other factor.
(x - 4) (x - 2) = 0 Divide both sides of the equation by (x - 4).
(x - 4) (x-4)
Remember that 0 divided by anything is 0 so 0 is also 0.
(x - 4)
Now you are left with (x-2) = 0, and to solve for x, add 2 to both sides of the equation and you will have your second answer.
x - 2 = 0
+ 2 +2
And you get x = 2, so your answers for this problem are x = 4 and x = 2.
10/09/15