Questions on Algebra: Systems of Linear Equations answered by real tutors!

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ok lets call the number of dimes and quarters d and q respectively.
:
dimes are worth 10cents and quarters are 25 cents so if you take the number of dimes(d) and multiply it by 10cents and the number of quarters (q) and multiply it by 25 cents then it will equal $7.30 right!!!!
:
.10d+.25q=7.30....eq 1.......and we know that the total number of coins(dimes and quarters) (d and q)=34
:
d+q=34....eq 2--->d=34-q
:
now we have two unknowns and two equations....take d's value from eq 2 and plug it into eq 1
:
.10(34-q)+.25q=7.3--->distribute
:
3.4-.1q+.25q=7.3---->combine like terms
:
.15q=3.90
:quarters
dimes

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Letters should be easier because you don't have to use much math to solve the equations.
A=P+Pr
a=p(1+r)
p=a/(1+r)

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I'll do the first two to get you started


# 1


Start with the given system of equations:





Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to solve for y.




So let's isolate y in the first equation

Start with the first equation


Subtract from both sides


Rearrange the equation


---------------------

Since , we can now replace each in the second equation with to solve for



Plug in into the second equation. In other words, replace each with . Notice we've eliminated the variables. So we now have a simple equation with one unknown.



Distribute the negative


Combine like terms on the left side


Add 7 to both sides


Combine like terms on the right side


Divide both sides by 7 to isolate x



Divide



-----------------First Answer------------------------------


So the first part of our answer is:





Since we know that we can plug it into the equation (remember we previously solved for in the first equation).



Start with the equation where was previously isolated.


Plug in


Multiply


Combine like terms



-----------------Second Answer------------------------------


So the second part of our answer is:









-----------------Summary------------------------------

So our answers are:

and

which form the point








Now let's graph the two equations (if you need help with graphing, check out this solver)


From the graph, we can see that the two equations intersect at . This visually verifies our answer.




graph of (red) and (green) and the intersection of the lines (blue circle).

---

# 2




Start with the given system of equations:



Multiply the both sides of the second equation by -2.


Distribute and multiply.


So we have the new system of equations:



Now add the equations together. You can do this by simply adding the two left sides and the two right sides separately like this:





Group like terms.


Combine like terms. Notice how the x terms cancel out.


Simplify.


Divide both sides by to isolate .


Reduce.


------------------------------------------------------------------


Now go back to the first equation.


Plug in .


Multiply.


Subtract from both sides.


Combine like terms on the right side.


Divide both sides by to isolate .


Reduce.


So our answer is and .


Which form the ordered pair .


This means that the system is consistent and independent.


Notice when we graph the equations, we see that they intersect at . So this visually verifies our answer.


Graph of (red) and (green)

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"the sum of two numbers is 180" translates to


"twice the first number increased by forty is equal to to twice the second number" translates to


Start with the second equation


Divide both sides by 2 to isolate y


Break up the fraction.


Reduce


So after isolating y, we get


Go back to the first equation


Plug in


Combine like terms on the left side.


Subtract from both sides.


Combine like terms on the right side.


Divide both sides by to isolate .


Reduce. So this is the first answer.


---------------------------------------------------


Go back to the isolated equation


Plug in


Add. This is the second answer.


=========================================================

Answer:

So the solutions are and which means that the two numbers are 80 and 100

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IF BOTH MATHODS GETS YOU THE CORRECT ANSWERS THEN YOU SHOULD CHOOSE THE SHORTEST METHOD. LESS CHANCE OF MAKING A MISTAKE IN THE MATH & YOU'RE USING WHOLE NUMBERS RATHER THAN FRACTIONS (WHICH CAN GET MESSY OR CONFUSING).

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Table of Contents:
Substitution
Elimination
Graphing




Jump to Top
Substitution:

Note: the first equation is the same as


Start with the second equation



Plug in


Distribute.


Combine like terms on the left side.


Subtract from both sides.


Combine like terms on the right side.


Divide both sides by to isolate .


Reduce. So this is the first part of the answer.


Go back to the first equation


Plug in .


Multiply and to get .


Combine like terms. This is the second part of the answer.


-------------------------------------------------------

Answer:

So the solutions are and


Which forms the ordered pair (0,6)

---

Jump to Top
Elimination:


Start with the first equation


Subtract 6 from both sides


Subtract "y" from both sides




Start with the given system of equations:



Multiply the both sides of the first equation by 4.


Distribute and multiply.


So we have the new system of equations:



Now add the equations together. You can do this by simply adding the two left sides and the two right sides separately like this:





Group like terms.


Combine like terms. Notice how the y terms cancel out.


Simplify.


Divide both sides by to isolate .


Reduce.


------------------------------------------------------------------


Now go back to the first equation.


Plug in .


Multiply.


Remove any zero terms.


Divide both sides by to isolate .


Reduce.


So our answers are and .


Which form the ordered pair . Note: this is the same answer as before.


This means that the system is consistent and independent.

---

Jump to Top

Graphing:


Start with the first equation


Rearrange the equation




Looking at we can see that the equation is in slope-intercept form where the slope is and the y-intercept is


Since this tells us that the y-intercept is .Remember the y-intercept is the point where the graph intersects with the y-axis

So we have one point




Now since the slope is comprised of the "rise" over the "run" this means


Also, because the slope is , this means:




which shows us that the rise is 2 and the run is 1. This means that to go from point to point, we can go up 2 and over 1



So starting at , go up 2 units


and to the right 1 unit to get to the next point



Now draw a line through these points to graph

So this is the graph of through the points and


----------------------------------------------------------


Now move onto the second equation


Subtract from both sides.


Rearrange the terms.


Divide both sides by to isolate y.


Break up the fraction.


Reduce.




Looking at we can see that the equation is in slope-intercept form where the slope is and the y-intercept is


Since this tells us that the y-intercept is .Remember the y-intercept is the point where the graph intersects with the y-axis

So we have one point




Now since the slope is comprised of the "rise" over the "run" this means


Also, because the slope is , this means:




which shows us that the rise is -3 and the run is 4. This means that to go from point to point, we can go down 3 and over 4



So starting at , go down 3 units


and to the right 4 units to get to the next point



Now draw a line through these points to graph

So this is the graph of through the points and


--------------------------------------------------------


Now let's graph the two equations together on the same coordinate system:



Graph of (red) and graph of (green)


Notice how the two lines intersect at the point (0,6). So this means that the solution is and (which confirms our previous answers)

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System of Equations with 2 variables


Easy Problem:




Solution: (7,3)

-----------------------------------------

Medium Problem:




Solution: (24/5, 2/5)


-----------------------------------------

Hard Problem:




Solution: (-253/106,187/106)



====================================================

System of Equations with 3 variables


Easy Problem:




Solution: (3,3,4)

-----------------------------------------



Medium Problem:




Solution: (43/10,-1/5,-9/5)

-----------------------------------------



Hard Problem:




Solution: (707/135, 5/54, -287/270)

-----------------------------------------

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"One number is 6 less than another." translates to


"The product of the number is 72" translates to


Start with the second equation


Plug in


Distribute.


Subtract 72 from both sides.


Notice we have a quadratic equation in the form of where , , and


Let's use the quadratic formula to solve for x


Start with the quadratic formula


Plug in , , and


Negate to get .


Square to get .


Multiply to get


Rewrite as


Add to to get


Multiply and to get .


Take the square root of to get .


or Break up the expression.


or Combine like terms.


or Simplify.


So the answers are or


This means that the numbers are:


12 and 6

OR...

-6 and -12

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how do you solve linear equaions with the subsitution method?
------------------------
You get one variable in terms of the other in one of the eqns, then substitute for it into the other eqn. The result is one eqn in one variable which you can then solve.
If there are more than 2 variables, you eliminate one at a time by substitution.
Look at the solutions for examples.

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This site gives you a very good tutorial on the subject:
http://www.purplemath.com/modules/systlin4.htm

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Problem:
Suppose $8,000 is invested at interest rate k, compounded continuously, and grows to $11,466.64 in 6 years.
a) Find the interest rate

 where , , 




Divide both sides by 


Take natural logs of both sides:


Use fact that 


Divide both sides by 6:

Use calculator:


We'll round that to  or 

---------------------

b) Find the exponential growth function

Substitute  for  and  for :



Your teacher may want you to write  in
functional notation as 

.  Ask your teacher.

-------------------

c) Find the balance after 10 years

Substitute  for  in

.  


  
    

--------

d) Find the doubling time.

For $8000 to double, it must become $16000,

So we substitute $16000 for A and solve for t

  



Divide both sides by 8000



Take natural logs of both sides:



Use fact that 



Divide both sides by .06:





or a little over  years.

Edwin

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Just multiply the part by 1 in the form of so that all of the denominators are the same, namely 4:

so that

(your original problem) becomes:



Now you can collect like terms in the numerators (because you can just add numerators anytime the denominators are equal).



Multiply both sides of the equation by 4



Add -2 to both sides



And finally divide both sides by -5



Check the answer by substituting for in the original equation and do the arithmetic to see if a true statement results. I'll leave this part as an exercise for the student.

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21=(4-5x)-(1-x)=4x-2(x-3) Too many equal signs
Here is my work/answer:
21+4-5x-1-x=4x-2x+6 That's better
4-5x-1=5x-2x+27
-1=-2x+31
30=-2x
x=-15
----------------
21+4-5x- 1+x =4x-2x+6 Change of sign
24-5x = 2x+6
18 = 7x
x = 18/7

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21=(4-5x)-(1-x)=4x-2(x-3) IF the first = sign is a + sign?????
Here is my work/answer:
21+4-5x-1-x=4x-2x+6
4-5x-1=5x-2x+27
-1=-2x+31
30=-2x
x=-15
---------------------------
Proof:
21+(4-5[-15])-(1-[-15])=4*-15-2(-15-3)
21+(4+75)-(1+15)=-60-2*-18
21+79-16=-60+36
84=-24 Obviously -15 isn't the correct answer.
-----------------------------------------------
21+4-5x+1{+x}=4x-2x+6
-5x+x-4x+2x=6-21-4-1
-6x=-20
x=-20/-6
x=10/3 ans.
Proof:
21+4-5*10/3+1+10/3=4*10/3-2*10/3+6
25-50/3+1+10/3=40/3-20/3+5
-40/3+26=20/3+6
-40/3-20/3=6-26
-60/3=-20
-20=-20

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4[3-3(x+1)]+1=2(-31-x)2x+31
Here's my work/answer:
12-12x-12+1=-62-2x+2x+31
-12x+1=-50-2x+2x=31
-12x+1=-50-2x+2x+19
-12x=-50-2x+2x+18
-12x=-32
x= 2 2/3
------------------
4[3-3(x+1)]+1=2(-31-x)2x+31
4[3-3x-3]+1 = -62-2x + 2x+31 (you're missing a plus sign above)
-12x + 1 = -31
-12x = -32
x = 8/3 = 2 2/3

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I was unsure how to write it in the first place, I'm sorry.
Its (7x/8) - 4=x The 7/8 is a fraction with the x on top with 7.
There is no need to say sorry. A lot of people do not know how to type math questions using their keyboard. I myself made a few errors when I started about 8 years ago.
Here is your question:
(7x/8) - 4 = x
Multiply each term on both sides of the fractional equation by 8 to remove the fraction given on the left side.
After doing so, we get:
7x - 32 = 8x....This is no longer a fractional equation. It is now called a linear equation. The fraction is gone!
We now solve for x.
-32 = 8x - 7x
-32 = x
===================
How do we know that x = -32 is right?
We plug that answer for x back into the original equation and simplify. If we get the same answer on both sides of the equation, then we will know that x = -32.
====================
THE PROVE:
7(-32)/8 - 4 = -32
-224/8 - 4 = -32
-28 - 4 = -32
-32 = -32...IT CHECKS!!!
Final answer: x = -32
I hope this helps.
Happy Thanksgiving.

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I think "substitution" is the proper label for the method.
Cheers,
Stan H.

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Start with the given system of equations:





In order to graph these equations, we must solve for y first.


Let's graph the first equation:


Start with the first equation.


Subtract from both sides.


Divide both sides by to isolate .


Rearrange the terms and simplify.


Now let's graph the equation:


Graph of .


-------------------------------------------------------------------


Now let's graph the second equation:


Start with the second equation.


Subtract from both sides.


Divide both sides by to isolate .


Rearrange the terms and simplify.


Now let's graph the equation:


Graph of .


-------------------------------------------------------------------


Now let's graph the two equations together:


Graph of (red). Graph of (green)


From the graph, we can see that the two lines intersect at the point . So the solution to the system of equations is . This tells us that the system of equations is consistent and independent.

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3x+2y=22
5x-4y=0
-------------
Elimination:
6x+4y = 44
5x-4y = 0
--------------
11x = 44
x = 4
---
3*4+2y = 22
2y = 10
y = 5
Ans: (4,5)
===========================
Substitution:
3x+2y=22
5x-4y=0
---------
x = (4/5)y
3(4/5)y + 2y = 22
(12/5+ 10/5)y = 22
(22/5)y = 22
(1/5)y = 1
y = 5
---
x = (4/5)5 = 4
Ans: (4,5)
===========================
Graphing:
3x+2y=22
5x-4y=0
------------
y= (-3/2)x + 11
y = (5/4)x

====================================
Cheers,
Stan H.

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Start with the given system of equations:





Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to solve for y.




So let's isolate y in the first equation

Start with the first equation


Subtract from both sides


Rearrange the equation


Divide both sides by


Break up the fraction


Reduce



---------------------

Since , we can now replace each in the second equation with to solve for



Plug in into the second equation. In other words, replace each with . Notice we've eliminated the variables. So we now have a simple equation with one unknown.



Distribute the negative


Combine like terms on the left side


Subtract 4 from both sides


Combine like terms on the right side


Since this equation is NEVER true for any x value, this means there are no solutions.


So the system is inconsistent

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Let X = number of girls in 1995
So, X+200%X= number of girls in 1998
.
Conclusion: There was an increase of 200% in 1998 since the count in 1995.
.
Proof:


.
Thank you,
Jojo

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is not defined when the denominator equals zero because division by zero is undefined.

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Hello, I just need help with how to solve this equation:
y-1/5x=-2
I know how to solve the problems, but only if there in simplified form already.. You don't have to give me the answer but just tell me how to simplfy it.
Thank you!!
------------------
There are 2 variables, or unknowns, but only 1 equation. You can graph it, it's a straight line, but there's no solution. Every point on the line will satisfy the 1 equation.
------------
You mention "simplify." You can express y in terms of x, or x in terms of y.
y = x/5 -2
------
x = 5y-10

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find the equation of the line with slope 8 and passing through(-5,-6).....
-------------------------
OK, I will.
y-y1 = m*(x-x1) (x1,y1) is (-5,-6), m is the slope
y+6 = 8*(x+5)
y+6 = 8x+40
y = 8x+34 (slope-intercept form)
8x - y = -34 (standard form)

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y-11=3x-6
y=3x-6+11
y=3x+5
(graph 300x200 pixels, x from -6 to 5, y from -10 to 10, 3x +5).

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You're on the right track.
Watch the parentheses though, they got you in trouble in the next line.


Multiply both sides by 2 to get rid of denominators.



Then from above

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"sum of a certain number and a second number is -42" ---->


"first number minus the second is 52" ---->




So we have the system of equations:




Add the equations together. You can do this by simply adding the two left sides and the two right sides separately like this:





Group like terms.


Combine like terms. Notice how the y terms cancel out.


Simplify.


Divide both sides by to isolate .


Reduce.


------------------------------------------------------------------


Now go back to the first equation.


Plug in .


Multiply.


Subtract from both sides.


Combine like terms on the right side.


So our answer is and .


Which form the ordered pair .


This means that the system is consistent and independent.

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lets call the numbers a and b
:
a+b=-42...eq 1
a-b=52....eq 2
using the elimination method to solve means we try to eliminate one of the variables and we can do that just by adding the 2 equation together(the b terms cancel out.
:
we end up with 2a=10---> divide by 2 and
:
to find b just plug a's value into either equation
:
5-b=52--->subtract 5 from each side and divide by -1

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--------------> A Line Equation
We follow Slope-Intercept form as we find the x & y intercepts:
It follows,
----> ;
f(y)=0:
, x- intercept
.
As we see below,
----> See x & y intercepts (-4,4). Also marked in "red" dots the slope ---> distance between points=1
.
Thank you,
Jojo

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# 1
Your first equation has two "x" variables. Is there a "y" term in the first equation?

---

# 2


Start with the given system of equations:



Multiply the both sides of the second equation by -5.


Distribute and multiply.


So we have the new system of equations:



Now add the equations together. You can do this by simply adding the two left sides and the two right sides separately like this:





Group like terms.


Combine like terms. Notice how the y terms cancel out.


Simplify.


Divide both sides by to isolate .


Reduce.


------------------------------------------------------------------


Now go back to the first equation.


Plug in .


Multiply.


Add to both sides.


Combine like terms on the right side.


Divide both sides by to isolate .


Reduce.


So our answer is and .


Which form the ordered pair .


This means that the system is consistent and independent.


Notice when we graph the equations, we see that they intersect at . So this visually verifies our answer.