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

 Page 6 / 13

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

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

#### Questions & Answers

explain and give four Example hyperbolic function
Lukman Reply
The denominator of a certain fraction is 9 more than the numerator. If 6 is added to both terms of the fraction, the value of the fraction becomes 2/3. Find the original fraction. 2. The sum of the least and greatest of 3 consecutive integers is 60. What are the valu
SABAL Reply
1. x + 6 2 -------------- = _ x + 9 + 6 3 x + 6 3 ----------- x -- (cross multiply) x + 15 2 3(x + 6) = 2(x + 15) 3x + 18 = 2x + 30 (-2x from both) x + 18 = 30 (-18 from both) x = 12 Test: 12 + 6 18 2 -------------- = --- = --- 12 + 9 + 6 27 3
Pawel
2. (x) + (x + 2) = 60 2x + 2 = 60 2x = 58 x = 29 29, 30, & 31
Pawel
Mark and Don are planning to sell each of their marble collections at a garage sale. If Don has 1 more than 3 times the number of marbles Mark has, how many does each boy have to sell if the total number of marbles is 113?
mariel Reply
Mark = x,. Don = 3x + 1 x + 3x + 1 = 113 4x = 112, x = 28 Mark = 28, Don = 85, 28 + 85 = 113
Pawel
how do I set up the problem?
Harshika Reply
what is a solution set?
Harshika
find the subring of gaussian integers?
Rofiqul
hello, I am happy to help!
Shirley Reply
please can go further on polynomials quadratic
Abdullahi
hi mam
Mark
I need quadratic equation link to Alpa Beta
Abdullahi Reply
find the value of 2x=32
Felix Reply
divide by 2 on each side of the equal sign to solve for x
corri
X=16
Michael
Want to review on complex number 1.What are complex number 2.How to solve complex number problems.
Beyan
yes i wantt to review
Mark
use the y -intercept and slope to sketch the graph of the equation y=6x
Only Reply
how do we prove the quadratic formular
Seidu Reply
please help me prove quadratic formula
Darius
hello, if you have a question about Algebra 2. I may be able to help. I am an Algebra 2 Teacher
Shirley Reply
thank you help me with how to prove the quadratic equation
Seidu
may God blessed u for that. Please I want u to help me in sets.
Opoku
what is math number
Tric Reply
4
Trista
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Sidiki Reply
can you teacch how to solve that🙏
Mark
Solve for the first variable in one of the equations, then substitute the result into the other equation. Point For: (6111,4111,−411)(6111,4111,-411) Equation Form: x=6111,y=4111,z=−411x=6111,y=4111,z=-411
Brenna
(61/11,41/11,−4/11)
Brenna
x=61/11 y=41/11 z=−4/11 x=61/11 y=41/11 z=-4/11
Brenna
Need help solving this problem (2/7)^-2
Simone Reply
x+2y-z=7
Sidiki
what is the coefficient of -4×
Mehri Reply
-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
Alfred Reply

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 By By By Richley Crapo By Dewey Compton By JavaChamp Team By OpenStax By Dan Ariely By Ann Schlosser By OpenStax By JavaChamp Team By Janet Forrester By Lakeima Roberts