Writing Systems Of Equations Homework Online

System-of-Equations Word Problems

In the past, I would have set this up by picking a variable for one of the groups (say, "c" for "children") and then use "(total) less (what I've already accounted for)" (in this case, "2200 – c") for the other group. Using a system of equations, however, allows me to use two different variables for the two different unknowns.

number of adults: a

number of children: c

With these variables, I can create equations for the totals they've given me:

total number: a + c = 2200

total income: 4a + 1.5c = 5050

Now I can solve the system for the number of adults and the number of children. I will solve the first equation for one of the variables, and then substitute the result into the other equation:

a = 2200 – c

4(2200 – c) + 1.5c = 5050

8800 – 4c + 1.5c = 5050

8800 – 2.5c = 5050

–2.5c = –3750

c = 1500

Now I can back-solve for the value of the other variable:

a = 2200 – (1500) = 700

I have values for my two variables. I can look back at my definitions for the variables to interpret these values. To answer the original question, there were:

1500 children and 700 adults.


You will probably start out with problems which, like the one above, seem very familiar. But you will then move on to more complicated problems.

  • The sum of the digits of a two-digit number is 7. When the digits are reversed, the number is increased by 27. Find the number.

The trick here is to work with the digits explicitly. I'll use "t" for the "tens" digit of the original number and "u" for the "units" (or "ones") digit. I then have:

The ten's digit stands for "ten times of this digit's value". Just as "26" is "10 times 2, plus 6 times 1", so also the two-digit number they've given me will be ten times the "tens" digit, plus one times the "units" digit. In other words:

original number: 10t + 1u

The new number has the values of the digits (represented by the variables) in reverse order. This gives me:

And this new number is twenty-seven more than the original number. The keyword "is" means "equals", so I get:

(new number) is (old number) increased by (twenty-seven)

10u + 1t = (10t + 1u) + 27

Now I have a system of equations that I can solve:

t + u = 7

10u + t = 10t + u + 27

First I'll simplify the second equation:

10u + t = 10t + u + 27

9u – 9t = 27

ut = 3

After reordering the variables in the first equation, I now have:

u + t = 7

ut = 3

Adding down, I get:

Then t = 2. Back-solving, this means that the original number was 25 and the new number (gotten by switching the digits) is 52. Since 52 – 25 = 27, this solution checks out.


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  • Find the equation of the parabola that passes through the points (–1, 9), (1, 5), and (2, 12).

Recalling that a parabola has a quadratic as its equation, I know that I am looking for an equation of the form ax2 + bx + c = y. Also, I know that points are of the form (xy). Practically speaking, this mean that, in each of these points, they have given me values for x and y that make the quadratic equation true. Plugging the three points in the general equation for a quadratic, I get a system of three equations, where the variables stand for the unknown coefficients of that quadratic:

a(–1)2 + b(–1) + c = 9

a(1)2 + b(1) + c = 5

a(2)2 + b(2) + c = 12

Simplifying the three equations, I get:

1ab + c = 9

1a + b + c = 5

4a + 2b + c = 12

I won't display the solving of this problem, but the result is that a = 3, b = –2, and c = 4, so the equation they're wanting is:


You may also see similar exercises referring to circles, using:

x2 + y2 + bx + cy + d = 0

...or other conics, though parabolas are the most common. Keep in mind that projectile problems (like shooting an arrow up in the air or dropping a penny from the roof of a tall building) are also parabola problems, using:

...where h0 is the original height, v0 is the initial velocity, s is the height at time t, usually measured in seconds, and g refers to gravity, being 9.8 if you're working in meters and 32 if you're working in feet).

All of these different permutations of the above example work the same way: Take the general equation for the curve, plug in the given points, and solve the resulting system of equations for the values of the coefficients. Warning: If you see an exercise of this sort in the homework, be advised that you may be expected to know the forms of the general equations (such as "ax2 + bx + c = y" for parabolas) on the next text.


URL: http://www.purplemath.com/modules/systprob.htm

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Many problems lend themselves to being solved with systems of linear equations. In "real life", these problems can be incredibly complex. This is one reason why linear algebra (the study of linear systems and related concepts) is its own branch of mathematics.

In your studies, however, you will generally be faced with much simpler problems. What follows are some typical examples.


  • The admission fee at a small fair is $1.50 for children and $4.00 for adults. On a certain day, 2200 people enter the fair and $5050 is collected. How many children and how many adults attended?

No answers ... yet!

System of Equations

A fishing rod and a reel cost a total of $110. The reel cost $100 more than the rod. What is the final cost of the rod and the reel?

Feb 22 | Kyna from Old Bridge, NJ | 0 Answers | 0 Votes

System Of EquationsWord ProblemNo answers ... yet!

At the hotdog stand it cost $18.39 for five hotdogs and six bags of chips if it cost $4.63 for one hotdog and two bags of chips how much is a hotdog

System of equations 

Feb 28 | Mckinley from Hartselle, AL | 0 Answers | 0 Votes

System Of EquationsNo answers ... yet!

How long will that take? How many of each type will there be?

system of equations answer to this question:  A daycare center in Salem currently has 9 assistant caregivers and 1 senior caregiver. Since demand is high, the owner is going to be hiring...

Feb 27 | Jenn from Kinsman, OH | 0 Answers | 0 Votes

System Of Equations

Bobosharif S.

Fort Myers, FL

i need help solving a system of linear equation by substitution 6x+y=30 , -9x+3y=-18

6x+y=30 , -9x+3y=-18

Feb 22 | Michael from San Antonio, TX | 1 Answer | 0 Votes

System Of Equations

Michael P.

Lenoir City, TN

System of Equations

1000 passengers ride the train during the morning hours in Hubbub City. A child's fare is $.50 and an adult fare is $1.50. $1300.00 in fares was collected this morning. How many adults and children...

Feb 21 | Kyna from Old Bridge, NJ | 1 Answer | 0 Votes

System Of Equations

Carol H.

Brownsburg, IN

I need some help ASAP Please.

This is the question----New York City is a popular field trip destination. This year the senior class at Franklin High School and the senior class at McKinney High School have both planned trips there...

Feb 12 | Bella from Broken Arrow, OK | 1 Answer | 0 Votes

MathSystem Of Equations

Dr Gulshan S.

Maple Valley, WA

North High School rented and filled 5 vans and 2 buses with 117 students.

This is the question----The senior classes at West High School and North High School planned separate trips to the state fair. The senior class at West High School rented and filled 10 vans and 6...

Feb 12 | Bella from Broken Arrow, OK | 1 Answer | 0 Votes

Math ProblemSystem Of EquationsNo answers ... yet!

How many lines do the two sizes of stanzas contain?

 Students in a poetry class are writing poems for their portfolios. The teacher wants them to write stanzas with certain numbers of lines each. Laura wrote 2 short stanzas and 10 long stanzas,...

Feb 20 | Maury from Las Vegas, NV | 0 Answers | 0 Votes

System Of Equations

Mikaila D.

Fort Lauderdale, FL

A club has $250. Buy (x) calculators for $8 each and sell them for $15 each. Buy (y) T-Shirts for $5 each and sell them for $18 each. Their goal is raise $365.

This is a homework question and I don't understand it. This question has to do with systems of equations, so you can solve using elimination or substitution. I am very confused so please help...

Jan 8 | Rabia from Easton, PA | 1 Answer | 1 Vote

System Of Equations

Bobosharif S.

Fort Myers, FL

System of equations

the sum of twice a number and triple a different number is equivalent to seventy two. But take half of that first number and increase it by eight and it will be equal to a third of that different...

Jan 30 | Briana from Reading, PA | 3 Answers | 0 Votes

System Of Equations

David W.

Round Rock, TX

16 nickels and dimes equal $1.35. How many nickels and dimes are there?

i don’t really have any additional details about my question so i’ll just fill up this space. 

Feb 8 | Payten from Waldorf, MD | 1 Answer | 0 Votes

System Of Equations

Mahdi G.

Edwardsville, IL

The width of a rectangle is eleven less than six times the length. -2 times the length minus three times the width -7. What is the width of the rectangle

 I’m trying to find the width and the length of the rectangle by using system of equation to represent my solution 

Feb 5 | Xaivion from East Saint Louis, IL | 1 Answer | 0 Votes

System Of Equations

Edward A.

Red Bluff, CA

write and solve the system of equations to find the number of packages of each type of cheese purchased

The Tran family is bringing 12 packages of cheese to the neighborhood picnic. Sliced cheese is $2.00 per package. Chunk cheese cost $1.50 per package. They spent $20.00 for cheese.

Feb 1 | Brianna from Sierra Vista, AZ | 1 Answer | 0 Votes

System Of Equations

David W.

Round Rock, TX

solve and create a system of equations

Ming sold 8 packages of chocolate chip cookie dough and 12 packages of gingerbread cookie dough for a total of $364. Carlos sold 1 package of chocolate chip cookie dough and 4 packages if gingerbread...

Jan 31 | Emma. from Hillsdale, MI | 1 Answer | 0 Votes

System Of Equations

Kenneth S.

Mesa, AZ

James has 5 and 10 pound weights, the make the total of 15 weights and in total weigh 150 pounds, find the solution.

the x and y can't be negative.

Jan 23 | Phillip from Vancouver, WA | 2 Answers | 0 Votes

System Of Equations

Mark M.

Bayport, NY

How do I solve this

4x + 5y =100  5x + 4y =62 If the system of equations above has solution (x,y) what is the value of x+y ?    A. 0    B. 9    C...

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Bobosharif S.

Fort Myers, FL

the larger of 2 numbers is 10 more than 3 times the smaller number. The larger number decreased by the smaller number is 80. What are the 2 numbers?

I just need to knoe the two numbers 

Jan 25 | Carol from Apex, NC | 1 Answer | 0 Votes

System Of Equations

David W.

Round Rock, TX

What is the solution to the system of equations?

What is the solution to the system of equations? 3x - 6y = -12 x - 2y = -8 (a) Use the substitution method to justify that the given system of equations has no solution...

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Bobosharif S.

Fort Myers, FL

there are 20 animals in a barn. some are chickens and some are horses. there are 64 legs in all how many of each are there

what is the system of the equation. show work please!

Feb 5 | Ann from Buford, GA | 1 Answer | -1 Votes

System Of Equations

David W.

Round Rock, TX

Rate A is a flat fee of $10 plus $10 per hour Rate B is $12 per hour What is the least number of hours she must tutor to make more money using rate B?

what are the equations for this system? 

Feb 5 | Sara from Buford, GA | 1 Answer | -1 Votes

System Of Equations

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