# 7.6 Solving systems with gaussian elimination  (Page 6/13)

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$\begin{array}{l}3x+2y=13\\ -x-9y+4z=53\\ 8x+5y+7z=80\end{array}$

For the following exercises, solve the system by Gaussian elimination.

No solutions

$\left(-1,-2\right)$

$\left(6,7\right)$

$\begin{array}{l}6x+2y=-4\\ 3x+4y=-17\end{array}$

$\left(3,2\right)$

$\begin{array}{l}-5x+8y=3\hfill \\ \text{\hspace{0.17em}}10x+6y=5\hfill \end{array}$

$\left(\frac{1}{5},\frac{1}{2}\right)$

$\left(x,\frac{4}{15}\left(5x+1\right)\right)$

$\begin{array}{l}11x+10y=43\\ 15x+20y=65\end{array}$

$\left(3,4\right)$

$\begin{array}{l}\begin{array}{l}\\ -1.06x-2.25y=5.51\end{array}\hfill \\ -5.03x-1.08y=5.40\hfill \end{array}$

$\begin{array}{l}\frac{3}{4}x-\frac{3}{5}y=4\\ \frac{1}{4}x+\frac{2}{3}y=1\end{array}$

$\left(\frac{196}{39},-\frac{5}{13}\right)$

$\begin{array}{l}\frac{1}{4}x-\frac{2}{3}y=-1\\ \frac{1}{2}x+\frac{1}{3}y=3\end{array}$

$\left(31,-42,87\right)$

$\left(\frac{21}{40},\frac{1}{20},\frac{9}{8}\right)$

$\left(\frac{18}{13},\frac{15}{13},-\frac{15}{13}\right)$

$\left(x,y,\frac{1}{2}\left(1-2x-3y\right)\right)$

$\left(x,-\frac{x}{2},-1\right)$

$\left(125,-25,0\right)$

$\begin{array}{l}\frac{1}{4}x-\frac{2}{3}z=-\frac{1}{2}\\ \frac{1}{5}x+\frac{1}{3}y=\frac{4}{7}\\ \frac{1}{5}y-\frac{1}{3}z=\frac{2}{9}\end{array}$

$\left(8,1,-2\right)$

## Extensions

For the following exercises, use Gaussian elimination to solve the system.

$\left(1,2,3\right)$

$\left(x,\frac{31}{28}-\frac{3x}{4},\frac{1}{28}\left(-7x-3\right)\right)$

No solutions exist.

## Real-world applications

For the following exercises, set up the augmented matrix that describes the situation, and solve for the desired solution.

Every day, a cupcake store sells 5,000 cupcakes in chocolate and vanilla flavors. If the chocolate flavor is 3 times as popular as the vanilla flavor, how many of each cupcake sell per day?

At a competing cupcake store, $4,520 worth of cupcakes are sold daily. The chocolate cupcakes cost$2.25 and the red velvet cupcakes cost $1.75. If the total number of cupcakes sold per day is 2,200, how many of each flavor are sold each day? 860 red velvet, 1,340 chocolate You invested$10,000 into two accounts: one that has simple 3% interest, the other with 2.5% interest. If your total interest payment after one year was $283.50, how much was in each account after the year passed? You invested$2,300 into account 1, and $2,700 into account 2. If the total amount of interest after one year is$254, and account 2 has 1.5 times the interest rate of account 1, what are the interest rates? Assume simple interest rates.

4% for account 1, 6% for account 2

Bikes’R’Us manufactures bikes, which sell for $250. It costs the manufacturer$180 per bike, plus a startup fee of $3,500. After how many bikes sold will the manufacturer break even? A major appliance store is considering purchasing vacuums from a small manufacturer. The store would be able to purchase the vacuums for$86 each, with a delivery fee of $9,200, regardless of how many vacuums are sold. If the store needs to start seeing a profit after 230 units are sold, how much should they charge for the vacuums?$126

The three most popular ice cream flavors are chocolate, strawberry, and vanilla, comprising 83% of the flavors sold at an ice cream shop. If vanilla sells 1% more than twice strawberry, and chocolate sells 11% more than vanilla, how much of the total ice cream consumption are the vanilla, chocolate, and strawberry flavors?

At an ice cream shop, three flavors are increasing in demand. Last year, banana, pumpkin, and rocky road ice cream made up 12% of total ice cream sales. This year, the same three ice creams made up 16.9% of ice cream sales. The rocky road sales doubled, the banana sales increased by 50%, and the pumpkin sales increased by 20%. If the rocky road ice cream had one less percent of sales than the banana ice cream, find out the percentage of ice cream sales each individual ice cream made last year.

Banana was 3%, pumpkin was 7%, and rocky road was 2%

A bag of mixed nuts contains cashews, pistachios, and almonds. There are 1,000 total nuts in the bag, and there are 100 less almonds than pistachios. The cashews weigh 3 g, pistachios weigh 4 g, and almonds weigh 5 g. If the bag weighs 3.7 kg, find out how many of each type of nut is in the bag.

A bag of mixed nuts contains cashews, pistachios, and almonds. Originally there were 900 nuts in the bag. 30% of the almonds, 20% of the cashews, and 10% of the pistachios were eaten, and now there are 770 nuts left in the bag. Originally, there were 100 more cashews than almonds. Figure out how many of each type of nut was in the bag to begin with.

100 almonds, 200 cashews, 600 pistachios

what is math number
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Need help solving this problem (2/7)^-2
x+2y-z=7
Sidiki
what is the coefficient of -4×
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
An investment account was opened with an initial deposit of \$9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
12, 17, 22.... 25th term
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Atone
hi
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
Vedant
find the 15th term of the geometric sequince whose first is 18 and last term of 387
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
hmm well what is the answer
Abhi
If f(x) = x-2 then, f(3) when 5f(x+1) 5((3-2)+1) 5(1+1) 5(2) 10
Augustine
how do they get the third part x = (32)5/4
make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be
AJ
how
Sheref
can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
hi
Ayuba
Hello
opoku
hi
Ali
greetings from Iran
Ali
salut. from Algeria
Bach
hi
Nharnhar