1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
V125BC [204]
3 years ago
12

In Exercises 40-43, for what value(s) of k, if any, will the systems have (a) no solution, (b) a unique solution, and (c) infini

tely many solutions?x - 2y +3z = 2x + y + z = k2x - y + 4z = k^2
Mathematics
1 answer:
svet-max [94.6K]3 years ago
5 0

Answer:

If k = −1 then the system has no solutions.

If k = 2 then the system has infinitely many solutions.

The system cannot have unique solution.

Step-by-step explanation:

We have the following system of equations

x - 2y +3z = 2\\x + y + z = k\\2x - y + 4z = k^2

The augmented matrix is

\left[\begin{array}{cccc}1&-2&3&2\\1&1&1&k\\2&-1&4&k^2\end{array}\right]

The reduction of this matrix to row-echelon form is outlined below.

R_2\rightarrow R_2-R_1

\left[\begin{array}{cccc}1&-2&3&2\\0&3&-2&k-2\\2&-1&4&k^2\end{array}\right]

R_3\rightarrow R_3-2R_1

\left[\begin{array}{cccc}1&-2&3&2\\0&3&-2&k-2\\0&3&-2&k^2-4\end{array}\right]

R_3\rightarrow R_3-R_2

\left[\begin{array}{cccc}1&-2&3&2\\0&3&-2&k-2\\0&0&0&k^2-k-2\end{array}\right]

The last row determines, if there are solutions or not. To be consistent, we must have k such that

k^2-k-2=0

\left(k+1\right)\left(k-2\right)=0\\k=-1,\:k=2

Case k = −1:

\left[\begin{array}{ccc|c}1&-2&3&2\\0&3&-2&-1-2\\0&0&0&(-1)^2-(-1)-2\end{array}\right] \rightarrow \left[\begin{array}{ccc|c}1&-2&3&2\\0&3&-2&-3\\0&0&0&-2\end{array}\right]

If k = −1 then the last equation becomes 0 = −2 which is impossible.Therefore, the system has no solutions.

Case k = 2:

\left[\begin{array}{ccc|c}1&-2&3&2\\0&3&-2&2-2\\0&0&0&(2)^2-(2)-2\end{array}\right] \rightarrow \left[\begin{array}{ccc|c}1&-2&3&2\\0&3&-2&0\\0&0&0&0\end{array}\right]

This gives the infinite many solution.

You might be interested in
What information does the ordered pair (3, 87) represent?
mafiozo [28]

<span>The 3 (x-coordinate) indicates that from the origin (0,0) in the cartesian plane, you have traveled four units to the right along the x-axis(the horizontal axis). </span>

<span>The 87 indicates that you traveled 87 units up in the y-axis(vertical axis), in line with the point 3 in the x-axis.This number is the y-coordinate, the second number in the ordered pair. </span>

8 0
3 years ago
It is believed that 11% of all Americans are left-handed. In a random sample of 100 students from a particular college with 1012
tia_tia [17]

Answer:

Step-by-step explanation:

We can calculate this confidence interval using the population proportion calculation. To do this we must find p' and q'

Where p' = 14/100= 0.14 (no of left handed sample promotion)

q' = 1-p' = 1-0.14= 0.86

Since the requested confidence level is CL = 0.98, then α = 1 – CL = 1 – 0.98 = 0.02/2= 0.01, z (0.01) = 2.326

Using p' - z alpha √(p'q'/n) for the lower interval - 0.14-2.326√(0.14*0.86/100)

= -2.186√0.00325

= -2.186*0.057

= 12.46%

Using p' + z alpha √(p'q'/n)

0.14+2.326√(0.14*0.86/100)

= 0.466*0.057

= 26.5%

Thus we estimate with 98% confidence that between 12% and 27% of all Americans are left handed.

6 0
3 years ago
At the beginning of the month, there were 80 ounces of peanut butter in the pantry. Since then, the family ate 0.3 of the peanut
givi [52]

Answer:

B i think

Step-by-step explanation:

you are supposed to multiply the 80*.3 and that will give you 24 and subtract from 80 but im not sure

sorry if i got this wrong

4 0
3 years ago
Pizza Planet is having a special;3 pizzas for 16.50. What is the unit rate
muminat
$5.50 per pizza /per unit
7 0
3 years ago
Find the length median of the trapezoid( 20 points and brainliest if answered)
Korolek [52]

Answer:

1. 36

2. 16

amdmkfefm

3 0
3 years ago
Other questions:
  • Mr. Walker asked his students to use the associative property to find an expression that is equivalent to (13 + 15 + 20) + (20 +
    13·1 answer
  • Calculators were purchased at $62 per dozen and sold for $38 for three calculators. Find the profit for selling 5 dozen calculat
    15·2 answers
  • Whats. The product of 1/3 + 12
    5·1 answer
  • A florist makes 215 bouquets in 5 days. How many does she make per day
    11·1 answer
  • Fraction and whole- Number Multiplication. 3/5 × 5​
    5·1 answer
  • <img src="https://tex.z-dn.net/?f=3x%20%2B%207%20%3D%2013" id="TexFormula1" title="3x + 7 = 13" alt="3x + 7 = 13" align="absmidd
    15·2 answers
  • The Newtown Telephone Company advertises that if you use their 10-10 service, you will always pay only 8¢ per minute, no matter
    10·1 answer
  • A store manager wishes to investigate whether there is a relationship between the type of promotion offered and the number of cu
    9·2 answers
  • If f, g, h
    8·1 answer
  • What is the area of this trapezoid? Enter your answer in the box.​
    14·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!