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

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The sum of two numbers is 30.
a + b = 30
:
The larger number is 5 more than four times the smaller one.
a = 4b + 5
:
Replace a in the 1st equation with (4b+5), find b
(4b+5) + b = 30
5b = 30 - 5
b = 25/5
b = 5, then a = 4(5)+5 = 25
:
"Find the product of the number."
25 * 5 = 125

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Let
Smaller number = y
Larger number = x

The sum of two numbers is 30
x+y=30..........(1)
The larger number is 5 more than four times the smaller one
x=4y+5..........(2)
Put the value of x from (2) to (1)
x+y=30..........(1)
4y+5+y=30
5y+5=30
5y=30-5
5y=25
5y/5=25/5
y=5
Put the value of y in (2)
x=4y+5..........(2)
x=4(5)+5
x=20+5
x=25


Smaller number = y = 5
Larger number = x = 25
Product of two numbers = 5*25 = 125

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ratio is 4:3
let the numbers be x & y (x>y)
x/y = 4/3
3x=4y
3x-4y=0.....................(1)
(x-4)=(y+8)
x-y = 8+4
x-y=12.....................(2)


3 x -4 y = 0 .............1
1 x -1 y = 12 .............2
Eliminate y
multiply (1)by 1
Multiply (2) by -4
3 x -4 y = 0
-4 x 4 y = -48
Add the two equations
-1 x = -48
/ -1
x = 48
plug value of x in (1)
3 x + -4 y = 0
144 + -4 y = 0
-4 y = 0 -144
-4 y = -144
y = 36
The numbers are 36 & 48

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Solve the 2X2 system.

John

My calculator said it, I believe it, that settles it

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Perform the indicated operation
7. (4-2i)(8+4i)
Simplify and write in a+bi form
10.i^38
Divide
13. 8+4i/7+3i
**
i=√-1
i^2=-1
i^3=-i
i^4=1
..
(4-2i)(8+4i)=32-16i+16i-8i^2=32+8=40
..
i^38=i^36*i^2=-1
..
8+4i/7+3i
multiply top and bottom by conjugate of denominator
(8+4i)(7-3i)/(7+3i)(7-3i)
(56+4i-12i^2)/(7^2-9i^2)
(56+4i+12)/(49+9)
(68+4i)/58
68/58+4i/58=34/29+2i/29

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is a vertical line through (4,0)
Any line parallel to this will also be a vertical
line. A vertical line that passes through (5,-2)
Also passes through (5,0) and has the equation
.

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The thrust of an airplane's engines produces a speed of 371 mph in still air. The wind velocity is given by (21,-35). In what direction should the airplane head to fly due south?
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Plot the point (21,-35) for the windspeed, label it A
Its length is sqrt(1666) =~ 40.81666
Its angle wrt the y-axis is arctan(21/35) =~ 30.964 degs
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Draw a line from A of length 371 (the plane's airspeed) from A southwest intersecting the y-axis.
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Use the Sine Law to find angle OBA
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A paperboy is riding at 9 ft/s on a bicycle and tosses a paper over his left shoulder at 25 ft/s. If the porch is 44 ft off the road, how far up the street should the paperboy release the paper to hit the porch
**
Time it takes for paper to reach porch=44/25 seconds
Distance up the street from porch needed to release the paper=9*44/25=15.84 ft

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No, they are consistent if they have a solution.
So we find what their solution must be by elimination:

2x + 4y = f
cx + dy = g

Multiply the first equation through by -d and the 
second equation through by 4

-2dx - 4dy = -df
 4cx + 4dy =  4g

(4c-2d)x = 4g-df

       x = 

Start over:

2x + 4y = f
cx + dy = g

Multiply the first equation through by c and the second equation through
by -2

 2cx + 4cy =  cf
-2cx - 2cy = -2g

(4c-2d)y = cf-26

       y = 
 
Notice that for both x and y, the denominators
are the same, 4c-2d

In order for those expresions for x and y to be
defined solution, their denominators must not be
zero, because division by 0 is not defined. But
division by any other number is defined.  Therefore 
the only requirement is that their denominator must 
not be equal to 0, so

       4c-2d ≠ 0

divide through by 2

        2c-d ≠ 0

          2c ≠ d

           d ≠ 2c

As long as d is not equal to 2c, there will be a solution for
any values of f and g.   

The answer to: "What can you say about the coefficients c and d?" is

d is not equal to 2c.

[If you have studied Cramer's rule, you can just say the determinant of
coefficients D must not equal 0.  The above was done assuming you have not
studied or were not allowed to use Cramer's rule.]

Edwin

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During one calendar year, a state trooper issued a total of 375 citations for warnings and speeding tickets. Of these, there were 37 more warnings than speeding tickets. How many warnings and how many speeding tickets were issued? Let w= warnings and s= speeding tickets
**
w+s=375
w=s+37
s+37+s=375
2s=375-37=338
s=338/2=169
w=s+37=206

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30 degrees is 30/360 which reduces to 1/12 of the whole circle (there are 360 degrees in a circle).
if 6pi is 1/12 of the circumference than the whole circumference can be found from 6pi*12 which gives us 72pi as my circumference.
Using the formula

36=r
I now know that my radius of the circle is 36.
Therefore my area of the circle is 1296 pi from the formula
Now that we have the area of the whole circle, I can just divide it by 12 to get the area of the sector. So my area is 108 pi as you said.

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Six $5 bills and one $10 = $40

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A woman can walk 2 mph faster down a trail than she can returning up the trail. It takes her 2 hours to get down and 4 hours to get back up. What is her speed down the trail?
=========================
Use the formula distance = speed x time
Let s = the speed down the trail
Then the speed up the trail = s-2
d = s*2 [down the trail]
d = (s-2)*4 [up the trail]
We can equate the distance down to the distance up:
d = (s-2)*4 = s*2
Solve for s:
4s - 8 = 2s
s = 4
So the speed down the trail is 4 mph

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y/7=(y-5)/2


2y=7(y-5)


2y=7y-35


2y-7y=-35


-5y=-35


y=(-35)/(-5)


y=7
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factor
-4x^3-36x
and factor
2x^2+8x-24
please help me!
--------------
Hint:
All the coefficients are even numbers.
Please make an effort !!!

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-4x+4y=12 solve for y
+4x. +4x
4y=4x+12
/4. /4. /4
y=x+3

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X Y
-3 7
-2 4
-1
0
1
2 -8
3
Use the first two pairs to create the equation: y = mx + b:
Assign as follows:
x1=-3, y1=7
x2=-2, y2=4
:
Find the slope m: = = -3
Use the point/slope form: y - y1 = m(x - x1)
y - 7 = -3(x - (-3))
y - 7 = -3x - 9
y = -3x - 9 + 7
y = -3x - 2: the equation we want
:
Substitute the given x values and find y, to complete the table
x=-1
y = -3(-1) - 2
y = 3 - 2
y = 1
the x/y pair -1,+1
x = 0
y = -3(0) - 2
y = -2
the pair 1,-2
:
the table now looks like this
X Y
-3 7
-2 4
-1 1
0 -2
1
2 -8
3
check our equation using x=2, then
you can do the same with x = 1 and x = 3, to finish the table

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What are the domain and range of the relation {(2, 2), (4, 2), (6, 2)}? Is the

The set of first coordinates is {2,4,6}, that its domain

The set of second coordinates is {2}, that its range.

relation a function? Choose yes or no.

The first coordinates are all different, so it's a function.
(It doesn't matter that the second coordinates are not different).

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Given f(x) = x² – 2x, find f(4).

Substitute 4 for x

      f(x) = x² – 2x
      f(4) = 4² – 2·4
      f(4) = 16 – 8
      f(4) = 8

Edwin

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i get x = -4/17
if i replace x in the original equation with -4/17, then i get:
1/3(x-7)+5=6x+4 becomes:
(1/3)*((-4/17)-7) + 5 = 6(-4/17) + 4
since am just confirming, i use my calculator to simplify this equation to:
2.588235294 = 2.588235294
this confirms the solution of x = (-4/17) is good.
looking at what you did, you were good up until:
1/3x-7/3+5=6x+4
your next statement looks like where you went wrong.
that statement is:
1/3x=6x+34/3
starting from your last good statement of:
1/3x-7/3+5=6x+4
you want to add 7/3 to both sides of the equation and you want to subtract 5 from both sides of the equation to get:
(1/3)x = 6x + 4 + 7/3 - 5
you can combine like terms to make this:
(1/3)x = 6x - 1 + 7/3
since -1 is equivalent to -3/3, you can combine like terms again to make this:
(1/3)x = 6x -3/3 + 7/3 which becomes:
(1/3)x = 6x + 4/3 ******* see note 1 below.
you now want to get all the x's over to the left side of the equation.
you do this by subtracting 6x from both sides of the equation to get:
(1/3)x - 6x = 4/3
since -6 is equivalent to -18, you can simplify this equation to become:
(1/3)x - (18/3)x = 4/3
you can now combine like terms to get:
(-17/3)x = 4/3
if you multiply both sides of this equation by -3/17), then you will get:
x = (4/3) * (-3/17) which you can simplify to:
x = (-12/51) which can be further simplified to:
x = (-4/17)
***** note 1:
this is where you went wrong.
you wound up with (1/3)x = 6x + 34/3
i wound up with (1/3)x = 6x + 4/3
i'm not sure what you did different, but see what i did and then see what you did that was different from that.

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Here is the answer: 3[4(y+2)-(y+5)]=3(3y+1)
3[4y+8-y-5]=9y+3
3[3y+3]=9y+3
9y+9=9y+3
9y+6=9y
0=6
This is a no solution problem, but the only mistake that I noticed, was that in the second stage, you didn't change the -(y+5) into -y-5. Otherwise, the answer would be correct.

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Hint: Use the idea that if |x| = k, for some positive number k, then x = k or x = -k. For example, if |x| = 3, then x = 3 or x = -3.

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See #554498
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email via the Thank You note, and I'll do it on paper, scan and send it.

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See #554498