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

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can you please help me solve: (((2x^2+x-6/2x+4))).... this is not an equation so
you can't solve it, but it's an expression that you can simplify:
;

:
Both the numerator and denominator can be factored to:

Note that you have (x+2) in both the numerator and denominator, cancel this and you have:

:
Did this make sense to you?

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complete: = ?/(x+1)(x-1)
:
Cross multiply
?(x-1) = 4x(x+1)(x-1)
:
? = 4x(x+1)
:
so we have:
=

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x^3+2x^2+3x-6/x-1
-----
Use synthetic division:
1)....1....2....3...-6
.......1....3....6..|..0
Quotient: x^2 + 3x + 6
Remainder: zero
===================
Cheers,
Stan H.

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Factor both numerator and denominator.

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can you please help me solve: (((n^3-27/n-3)))
---------------
This can't be solved, but it can be simplified.
Notice that in the NUM, both terms are the third power, n^3 and 3^3. Any binomial with odd exponents can be factored.
You can always just divide it out, too, with long division of synthetic division.
----------

The (n-3)'s cancel, so
it's

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Answer should equal 1 if like terms are combined.

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let x = one of the sides
let y = the base
---
base is a whole number that is 4 feet less than the sum of the two sides:
y = 2x-4
---
perimeter of the triangle is a whole number between 0 and 75.
---
maximum perimeter is 75.
equation for maximum perimeter would be:
2x + y = 75
since y = 2x-4, this equation becomes:
2x + 2x - 4 = 75
which becomes:
4x - 4 = 75
add 4 to both sides to get:
4x = 79
divide both sides by 4 to get:
x = 79/4 = 19.75
---
if x = 19.75, then substitution in the equation for the base and solve for y:
y = 2x-4
y = 2*19.75 - 4
y = 35.5
---
since the base has to be a whole number, 35.5 is no good.
the nearest integer is either 35 or 36.
i'll try 35 first.
---
y = 35
y = 2x-4
solving for x, we get:
x = (y+4)/2 = 39/2 = 19.5
---
perimeter = 2x + y = 39 + 35 = 74
this is a whole number and is less than or equal to 75 so it should be good.
----
let's try 36 next.
y = 36
y = 2x-4
solving for x, we get:
x = (y+4)/2 = 40/2 = 20
---
perimeter = 2x + y = 40 + 36 = 76
this is a whole number but it is not less than or equal to 75 so it is not good.
---
answer appears to be:
base = 35
side = 19.5
maximum perimeter = 74
---
the base is a whole number.
the perimeter is a whole number.
the base is equal to the sum of the sides plus 4.
the perimeter is greater than or equal to 0 and less than or equal to 75.
the perimeter cannot be greater than 74 since the base has to be a whole number and going to 36 for the base would result in a perimeter greater than 75 since the sides are in a fixed relationship to the base.
the problem does not state that the sides have to be integers so they can contain a fractional part.
-----
i believe your answer is:
base = 35
sides = 19.5 each
maximum perimeter = 74
---

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let x = the angle.
90 - x = the complement.
---
the angle is 25 degrees less than the measure of its complement.
x = 90 - x - 25
---
solve for x:
x = 90 - x - 25
add x to both sides of equation to get:
2x = 90 - 25 = 65
divide both sides by 2 to get:
x = 32.5
---
confirm this answer is good.
x = 90 - x - 25
32.5 = 90 - 32.5 - 25
32.5 = 90 - 57.5
32.5 = 32.5
answer is confirmed.
---
complement to the angle is 90 - 32.5 = 57.5
angle = 32.5 degrees
complement = 57.5 degrees
angle is smaller
answer is:
measure of the smaller angle is 32.5 degrees
---

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complemetary means 2 angles, when added together equal 90 degrees
:
let a and be be the complementary angles so a+b=90 ....let b=a-25....so plug b's value into the equation a+b=90
:
a+a-25=90
2a=115
:

so
:
check
57.5+32.5=90 and 57.5-32.5=25

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let x = the angle
180 - x = the supplement of the angle.
---
the angle measures 15 degrees more than twice its supplement.
x = 15 + (2 * (180-x))
simplify to get:
x = 15 + 360 - 2x
add x to both sides to get:
3x = 375
divide both sides by 3 to get:
x = 125 degrees
---
if x = 15 + 360 - 2x, then
x = 15 + 360 - 250 = 375 - 250 = 125 degrees.
this confirms the original equation so the answer is good.
---
the measure of the angle is 125 degrees.
this, however, is not what they asked.
---
the measure of the supplement is 180 - 125 degrees = 55 degrees
the answer is:
the measure of the supplement is 55 degrees.
---
to confirm again:
x = 15 + 2*55 = 15 + 110 = 125 degrees.
answer is confirmed.
---

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let the two angles be a and b.
:
a=2b+15.....eq 1
:
a+b=180.....eq 2
:
take a's value from eq 1 and plug it into eq 2
:
2b+15+b=180
:
3b=165
:
degrees
:
degrees

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What you have, written in a better way is

I'm not sure what level math you are at, but
here's how I'd do it:

multiply both sides by

multiply both sides by

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2/y=5 cross multiply.
5y=2
y=2/5
y=.4 ans.
-------------------
(1/3)/8 invert the denominator (8/1 becomes 1/8) & multiply.
1/3*1/8
1/24 ans.

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3* x - 1 = 5
-----------
3 1/3x - 1 1/6 = 5 1/2
(10/3)x - (7/6) = (11/2)
Multiply thru by 6 to get rid of the denominators:
20x - 7 = 33
20x = 40
x = 2
==============
Cheers,
Stan H.

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Note that you need to create new expressions using the information in the problem. "If the length is decreased by 10 in." is equivalent to saying 4x-10 since 4x represents the length. "and the width is increased by 2 in." is equivalent to saying x+2 since x represents the original width. Then, using the perimeter formula of P = 2W + 2L, we get that
80 = 2(x+2) + 2(4x-10)
Distributing:
80 = 2x + 4 + 8x - 20
Simplifying:
80 = 10x -16
Add 16 to both sides:
96 = 10x
Divide both sides by 10:
9.6 in = x
Thus the original width was 9.6 in and the original length was 4(9.6) = 38.4 in.

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0.4r - 1.2 = 0.3r + 0.6
:
Get rid of the decimals, multiply equation by 10
4r - 12 = 3r + 6
:
Subtract 3r from both sides:
4r - 3r - 12 = 6
:
Add 12 to both sides
r = 6 + 12
:
r = 18
:
:
Check solution in original equation, replace r with 18:
0.4(18) - 1.2 = 0.3(18) + 0.6
7.2 - 1.2 = 5.4 + .6
6 = 6; confirms our solution of r=18

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one of the method to solve system of equation is method of elimination.
4x+6y=38
10x+8y=60
lets eliminate x, to do so you have to multiply both equations first by 10 and second by -4 then you will get
40x+60y=380
-40x-32y=-240
as you see you got 40x in first equation and -40x in second - opposits, after adding first equation to the second x will be eliminated:
40x-40x+60y-32y=380-240
28y=140 / divide both sides by 28
y=5
to find x substitute y=5 in either first or second equation
4x+6*5=38
4x+30=38
4x=8
x=2

If you want eliminate y you will get same results:
4x+6y=38/*8
10x+8y=60/*-6
32x+48y=304
-60x-48y=-360
------------------
-28x=-56 / divide by -28
x=2
then 10*2+8y=60
8y=60-20
8y=40
y=5

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you have 2 equations in 2 unknowns.
you need to eliminate one of the unknowns so you can solve for the remaining one.
you do this by manipulating the equations so that when you add or subtract one equation from the other, the second unknown disappears.
---
here's how it's done.
your equations are:
4x + 6y = 38
10x + 8y = 60
---
if you multiply the first equation by 10, and you multiply the second equation by 4, you will be able to eliminate the x as follows:
---
first equation:
4x + 6y = 38
multiply both sides of the equation by 10 to get:
40x + 60y = 380
---
second equation:
10x + 8y = 60
multiply both sides of the equation by 4 to get:
40x + 32y = 240
---
your two equations are now:
40x + 60y = 380
40x + 32y = 240
---
subtract the second equation from the first to get:
0x + 28y = 140
which becomes:
28y = 140
divide both sides of this equation by 28 to get:
y = 5
---
now that you have solved for y, you can substitute in either equation to find x.
go back to your original equations:
4x + 6y = 38
10x + 8y = 60
---
take the second equation and substitute 5 for y to get:
10x + 8*5 = 60 which becomes 10x + 40 = 60
subtract 40 from both sides to get:
10x = 20
divide both sides by 10 to get:
x = 2
---
you should now have your answer:
x = 2
y = 5
---
to prove, substitute for x and y in the first equation:
4x + 6y = 38
4*2 + 6*5 = 38
8 + 30 = 38
38 = 38
values for x and y are good.
---
since you used the second equation to find x, you don't need to check that equation again, but we'll do it anyway to show you.
10x + 8y = 60
10*2 + 8*5 = 60
20 + 40 = 60
60 = 60
values for x and y are good.
---
you could have eliminated the y and solved for x first. i just eliminated the x and solved for y first. either way would have worked.

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let the measurements of our angles be a,b,and c
:
a=b+6........eq 1
c=3(a+b)-4...eq 2
a+b+c=180....eq 3
:
take a's value from eq 1 and plug it into eq 2.
:
c=3((b+6)+b)-4--->c=6b+18-4--->c=6b+14...revised eq 2
:
now take a's value from eq 1 and c's value from revised eq 2 and plug them into eq 3
:
b+6+b+6b+14=180
:
8b=160
:
degrees
:
degrees
:
degrees

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let the angles be x,2x,4x,and 8x
:
x+2x+4x+8x=360
:
15x=360
:

:
angles are

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let the two angles be a and b.
:
a=2b+15.....eq 1
:
a+b=180.....eq 2
:
take a's value from eq 1 and plug it into eq 2
:
2b+15+b=180
:
3b=165
:
degrees
:
degrees

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A=1/2bh
:
b+16=1/2b(6)
:
b+16=3b
:
2b=16
:

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          2x           +8
      --------------------- 
x - 4 )  2x2     - 32
         2x2 -8x
        --------------------
                        8x-32
                        8x-32
                        -----
                           0
                                 
Answer: 2x+8
:
or 
 

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==

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:

:

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Simplify:
Factor the numerator.
Now factor a 2 from the denominator.
Cancel the indicated factors.

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You can try this:
Factor the numerator.
Nothing will cancel so the fraction is already simplified.

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Given that (x+3) and (x-1) are both factors of the same expression x^3+ax^2-bx-a,find the values of the constants a and b.
Determine the third factor or the expression.
--------------------------------------------------
If x-1 is a factor, f(1) = 0
f(1) = 1^3 + a(1^2) - b(1) -a = 1 + a -b -a = 1-b = 0
Therefore b = 1
-------------------------
If x+3 is a factor, f(-3) = 0
f(-3) = (-3)^3 + a(-3)^2 - b(-3) -a = 0
-27 + 9a +3b -a = 0
Substitute b=1 and get
-27 +8a + 3 = 0
8a = 24
a = 3
===========
f(x) = x^3+ax^2-bx-a
f(x) = x^3 + 3x^2 -x - 3
-----------------------
Use synthetic division to find the 3rd factor:
1)....1....3....-1....-3
.......1......4...3...|..0
-3).....1......1..|..0
3rd factor: x + 1
=========================
Cheers,
Stan H.

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Hint: Factor

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(x-3/2)^3 x (1/x-3)^2
[(2x-3)/2]^3 x [(1-3x)/x]^2
= (1/8)[(2x)^3 - 3(2x)^2(3) + 3(2x)(3)^2 - (3)^3][(1/x)^3(1^3 -3(3x)+3(3x)^2-(3x)^3]
etc.
============
Cheers,
Stan H.

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can you please help me simplify:

Factor the left numerator and the right numerator
and denominator:



Put parentheses around the left denominator:



Indicate the multiplication of the numerators and denominators
all as one fraction:



Cancel the 's



Cancel the 's



Cancel the 's



All that's left is



Edwin

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can you please help me solve:

That is not an equation or an inequality, so it can't be
"solved".  It is an expression, and expressions cannot be
said to be "solved" but only "simplified" or "rewritten".
Only equations and equalities can be said to be "solved".     

To try to simplify it, we first try factoring the top:





or



But nothing cancels because we have - between a and b on
top and + between a and b on the bottom.  Are you sure you
typed all the signs right?

As you have it, it may be divided out by long division:

              a -  3b 
      ----------------
a + b )a2 - 2ab +  b2
       a2 +  ab
       --------
           -3ab +  b2
           -3ab - 3b2
           ----------
                  4b2

Then the expression may be rewritten as 

Or, you could rearrange the terms in descending order
of b instead of a, and get:

              b -  3a 
      ----------------
b + a )b2 - 2ab +  a2
       b2 +  ba
       --------
           -3ba +  a2
           -3ba - 3a2
           ----------
                  4a2

Then the expression can also be rewritten as 

So you can rewrite, not "solve", 



as either

, , or 

but expressions can only be simplified or rewritten.

Only equations and inequalities can be solved. 

Edwin

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:

:

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Complementary angles are angle such that the sum of their measure is 90 degrees.
If x is measure of the unknown angle, the measure of the complement of the angle is 90 - x.
The facts of the problem lead to the equation
So x (the measure of the unknown angle) = (90-x)-25
Manipulate the equation so that all terms involving the unknown "x" are on the same side of the equation.
So, 2x=90-25=65
Solve for x.
x = 65/2= 32.5 degrees.
I hope that this helps.

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complemetary means 2 angles, when added together equal 90 degrees
:
let a and be be the complementary angles so a+b=90 ....let b=a-25....so plug b's value into the equation a+b=90
:
a+a-25=90
2a=115
:

so
:
check
57.5+32.5=90 and 57.5-32.5=25

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The sum of the measure of the internal angles of a triangle is 180 degrees.
The equation that describes the situation in the problem is
x + x + (x + x -36) = 180 degrees where x is the measure of the angles with equal measure.
Adding like terms, we have 4x-36=180
So 4x=216 degrees
Solving for x, we have x = 216/4 = 54 degrees

So the measure of the 3 angles of the triangle are:
54, 54, 72 degrees
I hope that this helps.

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Let l be the length of the rectangle
Let w be the width of the rectangle.
From the description of the problem we can write the following equations:
l=w-3 (Equation #1)
2l + 2w = 54 (Equation #2)
Substitute equation #1 into Equation #2 and simplifify
2*(w-3)+2*w=54
2w-6+2w = 54
4w = 60
Solve for w
w = 60/4 = 15
Substitute w = 15 into Equation #2 to solve for l
2*l + 2(15) = 54
2*l + 30 = 54
2*l = 54-30 = 24
l = 12
Also, w-l =15-12=3 (agrees with equation #1)
I hope that this helps.

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how to do 2/3+9-1/24
------------
It's not clear what your expression is.
Is it (2/3) + 9 - (1/24)?
Is it 2/(3+9) - 1/24?
Or ((2/3) + 9 - 1)/24
or something else?

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Start with the given expression.


Plug in .


Square to get .


Multiply and to get .


Combine like terms.


So if

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Start with the given expression.


Factor out the GCF from the numerator


Factor 16 into


Highlight the common terms.


Cancel out the common terms.


Simplify


Distribute


So simplifies to


In other words, where

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I'm not sure what you mean by the "addition method" but I would solve the system of linear equations by attenpting to eliminate one of the unknowns.
Multiply 3x-2.5y=7.125 by 2.5 (Equation #1)
and multiply 2.5x-3y=7.3125 by 3 (Equation #2)
We have 2.5*(3x-2.5y*)=2.5*(7.125)
2.5*3*x - 2.5*2.5*y = 2.5*3*x-6.25*y = 17.8125 (#1)
and 3*2.5*x - 3*3*y = 3*7.3125* = 21.9375 (#2)
Subtract #2 from #1
-6.25*y + 9*y = 17.8125-21.9375
So 2.75*y = -4.125 or y = -4.125/2.75 = -1.5
Substitute -1.5 for y in equation #1
3*x - 2.5*(-1.5)=7.125
3*x + 3.75 = 7.125
3*x = 7.125 - 3.75 = 3.375
There x = 3.375/3 = 1.125
Check results
Using equation #1
3*(1.125)-2.5*(-1.5) = 3.375+3.75=7.125 (solution checks)

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1) -(6b-2)>O
-------------
Note: When an inequality is multiplied or divided on both sides by a
negative, the inequality sign is reversed.
-------------
If you multiply both sides of your problem be (-1) you get:
6b-2 < 0
6b < 2
b < 1/3
---------------------
2)-2(r-3)+7 >= 8
-2r + 6 + 7 >= 8
-2r + 13 >=8
-2r > = -5
2r <= 5
x <= 5/2
==================
Cheers,
Stan H.

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Start with the first equation.


Subtract the second equation from the first equation.


Distribute


Combine like terms.


So the answer is

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lets call the time spent at soup kitchen,trash pickup, and toy collecting
s, t, and c, respectively
:
c=4t...........eq 1
s=t-2..........eq 2
s+t+c=40.......eq 3
:
:
c=4t........eq 1
s+t+c=40....eq 3
:
multiply eq 1 by -1 to then add the two equations together thus eliminating the c terms, leaving us with s+t=-4t+40---->s+5t=40
:
s=t-2.....eq 2
s+5t=40...revised equation after eliminating c terms
:
multiply eq 2 by -1 and add the two equations together thus eliminating the s terms, leaving us with 5t=-t+2+40
:
6t=42
:
hours picking up trash
:
hours toy collecting
:
hours in the soup kitchen

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Start with the given expression.


Remove the parenthesis.


Group like terms.


Combine like terms.


So