Questions on Algebra: Expressions involving variables, substitution answered by real tutors!

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4w=3w+10
Subtrace 3w from both sides
w = 10

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y=4-3x
y=4-3*-3
y=4+9
y=13
(-3,13)

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Find the slope,if possible,of the line passing through each pair of points.
(2,-1)and(6,3)
-------------
The slope, m, is the diff in y over the diff in x
m = (-1-3)/(2-6)
m = -4/-4
m = 1
------
You try the other 2, email me to check your work or for assistance at
gsihoutx@aol.com

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90.3(2y-3)>21
180.6y-270.9>21
180.6y>21+270.9
180.6y>291.9
y>191.9/180.6
y>1.61628 ANS
PROOF:
LET y=1.62
90.3(2*1.62-3)>21
90.3(3.24-3)>21
90.3*.24>21
21.672>21

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90.3(2y-3)>21
180.6y-270.9>21
180.6y>21+270.9
180.6y>291.9
y>191.9/180.6
y>1.61628 ANS
PROOF:
LET y=1.62
90.3(2*1.62-3)>21
90.3(3.24-3)>21
90.3*.24>21
21.672>21

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5x+y=1 multiply by 2 & add.
10x-2y=7
10x+2y=2
--------------------
20x=9
x=9/20 ans.
5*9/20+y=1
45/20+y=1
y=1-45/20
y=20/20-45/20
y=-25/20
y=-5/4 ans.
Proof:
10*9/20-2*-5/4=7
90/20+10/4=7
4.5+2.5=7
7=7

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+1x-2y+3z=37
-2x-3y=2z=16 Too many equal signs
-2x-1y+1z=-2
------------

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Invert the denominator and multiply:


Now, FACTOR, FACTOR, FACTOR:
*

Strangely enough, the first denominator does NOT factor (are you sure you didn't copy this wrong???), and no factors divide out. The final answer here is really UGLY, and I doubt if it is correct.


Dr. Robert J. Rapalje, Retired
Seminole State College of Florida
Altamonte Springs Campus

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The variable is simply the unknown number. In the case of 6y-3=6y+8, the variable is the letter y. The letter y simply stands for some unknown number.

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With this information all we can do is move 9x to the right side of the equation. So

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LENGTH CAN'T BE 1/2 LENGTH.

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(t-3)^2=8
t^2-6t+9=8
t^2-6t+9-8=0
t^2-6t+1=0

t=(6+-sqrt[-6^2-4*1*1])/2*1
t=(6+-sqrt[36-4])/2
t=(6+-sqrt32)/2
t=(6+-5.6568)/2
t=11.6568/2
t=5.828 ans.
t=(6-5.6568)/2
t=.3432/2
t=.1716 ans.

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LET THE MAN'S AGE=4X
LET THE SON'S AGE-X
(4X+4)(X+4)=520
4X^2+4X+16X+16=520
4X^2+20X+16-520=0
4X^2+20X-504=0
4(X^2+5X-126)=0
4(X-9)(X+14)=0
X-9=0
X=9 AGE OF THE SON.
4*9=36 AGE OF THE DAD.
PROOF:
(4*9+4)(9+4)=520
(36+4)(13)=520
40*13=520

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5x - 6x = -17
3x - 4y + 5z = -1
2z = -6

looks like you have 3 equations that have to be solved simultaneously.

looks like your third equation has already solved for z so we can start there.

2z = -6 becomes z = -3 when you divided both sides of the equation by 2.

z = -3 is the value of one of your variables.

you are left with:

5x - 6x = -17
3x - 4y + 5z = -1

that first equation looks like it's all in x so we can solve for x from that equation.

5x - 6x = -17 becomes -x = -17 after you combine like terms.

multiply both sides of this equation by -1 to get:

x = 17

you now have two values out of the 3.

you have:

x = 17
z = -3

you have only one equation left to solve.

that is:

3x - 4y + 5z = -1

replace x with 17 and z with -3 in that equation to get:

3*(17) - 4*y + 5*(-3) = -1

this becomes:

51 - 4*y - 15 = -1

combine like terms to get:

36 - 4*y = -1

add 4*y to both sides of the equation and add 1 to both sides of the equation to get:

36 + 1 = 4*y

combine like terms to get:

37 = 4*y

divide both sides of the equation by 4 to get:

9.25 = y

this is the same as:

y = 9.25

the answer to your equation is:

x = 17
y = 9.25
z = -3

substitute in your original equations to confirm these answers are good.

5x - 6x = -17
3x - 4y + 5z = -1
2z = -6

these become:

5*(17) - 6*(17) = -17
3*(17) - 4*(9.25) + 5*(-3) = -1
2*(-3) = -6

simplify and combine like terms to get:

-17 = -17
51 - 37 - 15 = -1
-6 = -6

simplify further to get:

-17 = -17
-1 = -1
-6 = -6

all equations confirm so the answer is good.

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y = k*x

if y = 38 and x = 114, this becomes:

38 = k*114

k = 38/114

k = (1/3)

This is saying that y = (1/3) * x

38 = (1/3) * 114

38 = 38 confirming the value for k is good.

they are directly proportional which means that an increase in one yields an increase in the other by the stated proportion.

suppose y = 1

formula states that 1 = (1/3)*x which makes x = 3.

An increase of 1 in y will result in an increase of 3 in x.

2 = (1/3)*x makes x = 6

y increased by 1
x increased by 3

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Use the distributive rule.
-(d + 91) = -d - 91
They both mean the same thing.

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how to solve using linear combination's
2x-y=8
x+y=1
-------------------------
Multiply thru the 2nd equation by 2 to get:
2x-y = 8
2x+2y = 2
------------------------------
Subtract the 1st from the 2nd Eq:
3y = -6
y = -2
------------------
Substitute into x+y = 1 to solve for "x":
x + -2 = 1
x = 3
---------------------------
Cheers,
Stan H.

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D=RT
D=100T
D=110(T-.5)
100T=110(T-.5
100T=110T-55
100T-110T=-55
-10T=-55
T=-55/-10
T=5.5 HOURS AFTER 4:15 TRAIN B WILL CATCH TRAIN A.
PROOF:
100*5.5=110(5.5-.5)
550=110*5
550=550

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Your current employer pays you a flat rate to deliver packages. You discover that a competitor pays its employees hourly in addition to paying them a flat rate per package.
Question:
Develop an algebraic equation with clearly defined variables to represent the cost/yield of each option.
my first attempt:
X= packages delivered Y = hours worked
a. .75x = .05x+8(15)
b. .75(200) = .05(200) + 8y
---------------------------------------
Your employer:
If flat rate is $0.75, and the number of packages you deliver is "x"
and your pay is A(x),
A(x) = 0.75x
-----------------------
Other employer:
If his hourly rate is "0.25" per h hours, and his package rate is "0.15" per
package, and the number of packages delivered is "x" and the pay is B(x)
B(x) = 0.25h + 0.15x
------------------------
Cheers,
Stan H.

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Hello,
First, let's find the slope of the line using the slope formula (y-y)/(x-x)
(-5,-1) and ( -2,-8) It will look like this:
-1-(-8)/-5-(-2) will give us:
7/-3 or -7/3 Now we can use the point-slope formula which is:
y-y=m(x-x) Plug in either of the points and we have: (Note: m is slope)
y-(-1)=-7/3(x-(-5) Simpilfy this to:
y+1= -7/3x - 35/3 Subtract 1 both sides which gives us:
y=-7/3(x) - 38/3 This can be rewritten as:
y = -2 1/3(x) - 12 2/3
RJ Toftness
www.math-unlock.com

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Hello,
Let's first multiply the first equation by -1 (note you could multiply the first or second equation)
(-1)x (-1)(- 9y) = (-1)101 This will then be:
-x +9y = -101 Now we can add it to the second equation.
x - 7y = 75 Notice the x cancels out leaving us with:
2y = -26 Divide both sides by 2 which gives us:
y= -13 Plug this into the original equation to find x.
x = -16
RJ Toftness
www.math-unlock.com

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2p+p=1/4*2p+150
2.5p=150
p=60

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Too many equal signs.
Cheers,
Stan H.

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please help me solve
a/b=c solve for b
---
Invert both sides to get:
b/a = 1/c
Multiply both sides by a to get:
b = a/c
---------------------
h-4/j=k solve for j
---
Invert both sides to get:
j/(h-4) = 1/k
Multiply both sides by h-4
j = (h-4)/k
=============================
Cheers,
Stan H.

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Now that is a very handsome little 2-variable linear equation. What was it you wanted to do with it besides share it with us?

John

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Well you want to get n on the one side of the equation.
So first step would be to take 4n away from each side to give
2n-42 = 0
Can you work out the next step?
Yes you are right add 42 to each side to give:
2n = 42
Final step and solution?

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IF THIS IS A LINEAR EQUATION THEN:
two x+ one third y = 7
2X+Y/3=7
Y/3=-2X+7 MULTIPLY BY 3.
Y=-2*3X+3*7
Y=-6X+21
(graph 300x200 pixels, x from -6 to 5, y from -10 to 20, -6x +21).

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Here is your equation:
4(x + 3) = 16
Can you solve for x now?

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We need to isolate m to find its value, so...
Steps:
a. add 3 to both sides of the equation
b. multiply by 3 on both sides of the equation to find value of m
.
Problem:




One more step and you're done. I'll let you finish it. :)

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300 = 16t^2
t^2 = 300/16
t = sqrt(75/4)
t =~ 4.33 seconds

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(d + 1)(d - 1 + 6) d = 2
Replace every d with 2 and do the math.
(2 + 1)(2 - 1 + 6) = 3 times 7 = 14

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Since the reciprocal of x is 1/x, y = 1/x. Substituting this into the expression for y we get:

Now we simplify. We'll start with the second term by eliminating the fractions within the "big" fraction. This is done by multiplying the numerator and denominator of the "big" fraction by the Lowest Common Denominator (LCD) of the "little" fractions. Since there is only one "little" fraction, finding the LCD is simple. It is x.



Not only have we eliminated the fraction with a fraction, we have coincidentally gotten the denominators to match (since 1+x=x+1 by the Commutative Property). So we can go ahead and add the two fractions:

And finally, as usual with answers that are fractions, we try to reduce the fraction. This is very easy and obvious, I hope.


which simplifies to

The x's have disappeared. This means that is equal to 1 no matter what number you use for x (except x cannot be zero because zero does not have a reciprocal).

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xy-8=t
xy=t+8
x=(t+8)/y
.
Ed

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36x-12y-3z=81
-3z=-36x+12y+81
z=12x-4y-27
.
Ed

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That is a lot of info, so please visit this web page:
http://www.purplemath.com/modules/simpexpo.htm

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Take and solve for 'b' to get . Now plug this into to get . From here, you have a quadratic which can be solved by the quadratic formula.

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This equation is actually the conversion of Fahrenheit degrees into Celcius. The problem is now asking you to find the conversion equation of Celcius degrees into Fahrenheit. You will see them again in future classes and your life, so it's good to learn and remember them. Here we go:
.

.
We need to isolate F, so first, let's get rid of 5/9 by multiplying its reciprocal on both sides of the equation... its reciprocal is 9/5.

Now, let's add 32 to both sides to leave F alone.

Or you can state it as:

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Steps:
a. leave the first equation alone for now.
b. solve for one of the variables using the second equation. Let's solve for m.
c. then use the value of m and substitute it in the first equation to solve for the other variable.
Problem:


Now use this value and plug it in the first equation.






Now, we look for the exact value of m.




Your values are: n=-7 and m=1
=================================
If you need to check your answers (some professors do ask you to do it), just substitute these two values in the original equations to comfirm they add up correctly. This is good to do it to ensure you didn't miss any negatives when you worked out your problem.

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6*3/(6+3)=18/9=2 hours to finish the job working together.

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I need help plz.. Solve by the substitution method. What is the solution?
3x - 2y =-6
5x + 24 =y
------------------
Subtitute for "y" in the 1st equation and solve for "x".
---
3x - 2(5x+24) = -6
3x - 10x - 48 = -6
-7x = 42
x = -6
-----
Substitute into y = 5x+24 to solve for "y":
y = 5*-6 + 24 = -6
--------------------------
Cheers,
Stan H.

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your company produces two models of bikes Model A and Model B Model A takes 2 hours to assemble and Model B takes 3 hours to assemble. Model A cost $25 to make each bike where Model B costs $30 to make per bike. If your company has 34 hours and $350 per day for these two bikes, how many of each can be made in a day? Solve equation using substitution or elimination. Explain how to check your solution for both equations.
:
Let a = no. of Model A bikes
Let b = no. of B Model
:
The hrs constraint
2a + 3b = 34
:
the cost constraint
25a + 30b = 350
:
Multiply the 1st equation by 10, subtract from the above equation
25a + 30b = 350
20a + 30b = 340
------------------subtraction eliminates b, find a
5a = 10
a = 2 Model A bikes
then using the 1st equation
2(2) + 3b = 34
4 + 3b = 34
3b = 34 - 4
3b = 30
b = 10 model B bikes
;
:
Check in the cost equation
25(2) + 30(10) = 350

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I think I have exactly what you may need to help you with the Laws of Exponents. Please see my own website for my "non-traditional" explanation. Do a "Bing" or "Google" search for my last name "Rapalje". Near the top of the search list, look for "Rapalje Homepage." Near the top of my Homepage, look for the link "Basic, Intermediate and College Algebra: One Step at a Time." Select "Basic Algebra" and look in Chapter 2 for "Section 2.12 Positive Exponents" and "Section 2.13 Negative and Zero Exponents." Also, please see the sections from the "MATH IN LIVING COLOR" pages that correspond to these sections. Other sections from my curriculum may also be helpful. It's all free. If you like my website, please recommend it to family and friends who can use a more informal explanation of math. Feel free to send me an Email for help finding a particular topic.

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida
Altamonte Springs Campus

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12a to the 3rd power.. over 4b to the 2nd power.

All you can do here is cancel the 4 into the 12

a & b are unlike terms, so that's all you can do with it
:
(16a)to the 2nd power over (16a)cube.

First 16/16 is one so they cancel and you have

When you divide like terms you subtract the exponents, in this one
= which is just
:
The other rules are simple
when you multiply like terms you add the exponent ex: a^5 * a^2 = a^7
When you raise one power to another power you multiply exponents: ex (a^4)^2 = a^8