Questions on Algebra: Linear Algebra (NOT Linear Equations) answered by real tutors!

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Is that all one problem?

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You didn't provide the graph, so I can only answer your questions in either very general or procedural terms.

Look at the value of the point on the graph where

Look at the value where on the graph.

Write down the ordered pairs that represent the end points of each segment of the graph, (,) = (age, amount)

For each pair of endpoints that define a segment, use the slope formula to calculate the slope of that segment:



where

and

are the coordinates of the endpoints. The highest slopes give the greatest benefit for waiting that particular year.

People wait until 70 to draw their social security because they are betting that they will live long enough to accumulate more benefits than would accrue if you took the lower benefit earlier.

For example, if the annual benefit is 18000 at age 65 and the annual benefit is 24000 at age 70 and

You live to be 80 years old: You will accumulate 15 times $18,000 = $270,000 if you take the benefit at 65, but only 10 times 24,000 = $240,000 if you take the benefit at 70. So if you are only going to live to be 80, it is better to start taking the money earlier

At age 85, with the given amounts, you break even. That is, it doesn't matter which you do, so you might as well enjoy the additional years of retirement -- leave early.

But if you are planning to live beyond age 85, you start to make money by delaying retirement to age 70.


John

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There are many methods for solving systems, and
I don't know which way you are studying. I don't want 
to waste time solving them all the different 
possible ways which you aren't studying, so I'll just
give the solution:

(x,y,z) = (-5,-2,3)

If you want us to help you with solving it, please
tell us which of the many ways you are studying, probably
one of these:

1. Substitution
2. Elimination
3. Cramer's rule (determinants)
4. Triangular method with back substitution
5. Partial Gaussian elimination with matrices and back substitution 
6. Complete Gaussian elimination with matrices
7. The inverse matrix method, sometimes called the AX=B method

Edwin

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caramels costing 1.75 per pound, are mixed with cream chocolate costing 2 per
pound to make a 5 pound mixture that will be sold at 1.90 per pound..
how many pounds of each are needed?
:
Let x = amt of stuff costing $2 a lb
then
(5-x) = amt of stuff costing $1.75 a lb
:
2x + 1.75(5-x) = 1.90(5)
solve for x

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Here's a great web page that explains it all and presents some examples as well.

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A company designs and sells two types of rings: the VIP and the SST.
The company can produce up to 24 rings each day using up to 60 total hours of labor. It takes 3 hours to make one VIP ring, and 2 hours to make one SST ring. How many of each type of ring should be made daily in order to maximize profit, if profit on a VIP ring is $30 and profit on an SST ring is $40? What is the profit?
So far I have come up with this not sure if its correct?
x=# of rings to VIP
y=# of rings to SST
3x+2y< or equal to 60
x+y< or equal to 24
Profit = 30x+40y
-----------------------
Graphs:
3x+2y <= 60
y <= (-3/2)x + 30
--------
x+y <= 24
y <= -x+24
--------

-----------
Find the intersection of the two boundary lines:
-x+24 = (-3/2)x+30
(1/2)x = 6
x = 12 so y = 12
========================
Check (0,0),(0,24),(20,0), (12,12) in the profit equation to determine the max pair.
-------------------------
Profit = 30x+40y
P(0,0)=0 ; P(0,24)=40*24=960;P(20,0)=30*20=600;P(12,12)=30*12+40*12 = 840
---
Max comes from x = 0, y = 24
====================================
Cheers,
Stan H.

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When 0.5 cm was planed off each of the six faces of a wooden cube, its volume decreased by 169 cm^3. Find its new volume.
--------------
Each side is decreased by 1 cm, 0.5 on each face.
(s+1)^3 = s^3 + 169
s^3 + 3s^2 + 3s + 1 = s^3 + 169
3s^2 + 3s - 168 = 0
s^2 + s - 56 = 0
(s + 8)*(s - 7) = 0
s = 7 cm
V = 7^3
V = 343 cc
----------
343 + 169 = 512 = 8^3

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h(x)=23
-----------
It's linear, and parallel to the x-axis.
For any value of x, h = 23.

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Your objective function is not quite right. Since you want to maximize profit, the profit on a small basket is 16 - 10 = 6, and on a large basket is 25 - 15 = 10, so your objective function needs to be:



You won't sell more than 100 baskets, per Pres. estimate so:



You can't spend more than $1200 procuring the baskets, and smalls cost 10 and larges cost 15, so:



You can't sell less than zero of either kind of basket, so:



and



And, since it is highly unlikely that anyone will buy a fractional part of a basket of either size both quantities must be integers:



Your area of feasibility, based on the above constraints is integer values where the ordered pair (x, y) is in the region bounded by the quadrilateral with vertices (0, 0), (100,0), (60, 80), and (0,80). You need to graph all of the constraints and solve the system:





to find these points for yourself.

There is an operations research theorem that says the optimum point will be at a vertex of the area of feasibility. So, just plug the vertex values into the objective function and see which one gives you the biggest answer.

John

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Let X = the length of the rod
Let Y = the length of the leader
Equation 1:
Equation 2:
Using equation 1, plug (X - 2) into equation 2 for Y

Rewrite the equation
Subtract 16 form both sides
Then you need to use the quadratic equation

Solved by pluggable solver: SOLVE quadratic equation with variable

Quadratic equation (in our case ) has the following solutons: For these solutions to exist, the discriminant should not be a negative number. First, we need to compute the discriminant : . Discriminant d=68 is greater than zero. That means that there are two solutions: . Quadratic expression can be factored: Again, the answer is: 5.12310562561766, -3.12310562561766. Here's your graph:

Since you can not have a negative length, the 5.123 is the only rational answer.
But remember that 5.123 is the length of the rod.
To find the length of the leader, plug 5.123 into equation 1 for X
Equation 1:


The leader = 3.123ft

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Let represent the width. Then the length must be . Since the area is the length times the width,



Anything equal to or larger than the positive root of:




John

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This is a substitution problem.
-------
Given:
f(X) = 8 + X
g(X) = X - 5
d(X) = f(X) - g(X), find d(X)
All you need to do now is plug (8 + X) in for f(X) and plug (X - 5) in for g(X) into the d(X) equation.

rewrite the equation
Add like terms

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If you meant to say that he scored three 3-point shots, then it works out this way.

Three 3-point shots is 3 shots worth 9 points, so the field goals and free throws have to account for 17 of the 20 shots and 25 of the 34 points.

Let represent the number of field goals and the number of free throws. Then is the number of field goal points and is the number of free throw points.





Multiply the first equation by -1:



Add this new equation to the original 2nd equation:



So, 8 field goals, and 17 - 8 = 9 free throws.

Check: 3 three-pointers for 9 points, 8 field goals for 16 points, and 9 free throws for 9 points. 3 + 8 + 9 = 20 and 9 + 16 + 9 = 34. Checks.

John

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The distance formula between two points is:

Now all you need to do is substitute values and solve algebraically.

If you raise both sides of the eqn by the power of 2, you will get:




Now take the square root of both sides, you get:

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Complete the square. Step 1, divide the coefficient on the 1st degree term by 2 and square the result:

Add and subtract this value in the RHS



The first three terms in the RHS are now a perfect square:



Vertex at


John

What is the point slope form with the table 
(X) 3,5, 7, 9
(Y)-6,2,10,18

What is the point slope form with the table 
(X) 3,5, 7, 9
(Y)-6,2,10,18

Let's plot those points and see if they are in a straight line:

 

They look like they do, judging from the above. We draw a
line through them:

 

We can pick any two points of the four points 
to find the slope of the line.  We'll get the
same slope no matter which two of the points
we pick.  Let's pick (5,2) and {9,18).

To find the slope we use the slope formula:



where , , , 



To find the equation, we use the point slope form:





and we solve for y and simplify to get the 

slope intercept form:





Edwin