Makayla Boyd

2022-06-21

Correct me if I am wrong. Find the value(s) of the constant k such that the system of linear equations

$\{\begin{array}{l}x+2y=1\\ {k}^{2}x-2ky=k+2\end{array}$

has:

1. No solution

2. An infinite number of solutions

3. Exactly one solution

Answer:

so the first step is to get row reduction form, which is:

from $\left[\begin{array}{cc}1& 2\\ {k}^{2}& -2k\end{array}\right]$,

to $\left[\begin{array}{cc}1& 2\\ 0& -2k+2{k}^{2}\end{array}\right]$

$\{\begin{array}{l}x+2y=1\\ {k}^{2}x-2ky=k+2\end{array}$

has:

1. No solution

2. An infinite number of solutions

3. Exactly one solution

Answer:

so the first step is to get row reduction form, which is:

from $\left[\begin{array}{cc}1& 2\\ {k}^{2}& -2k\end{array}\right]$,

to $\left[\begin{array}{cc}1& 2\\ 0& -2k+2{k}^{2}\end{array}\right]$

Jayce Bates

Beginner2022-06-22Added 18 answers

You row reduction is wrong. We get

$\begin{array}{ccc}1& 2& 2\\ 0& -2{k}^{2}-2k& -{k}^{2}+k+2\end{array}$

which is equivalent to

$\begin{array}{ccc}1& 2& 2\\ 0& 2k(k+1)& (k+1)(k-2)\end{array}$

From here we see that there is no solution iff $k=0$, an infinite number of solutions iff $k=-1$ and else there is exactly one solution.

$\begin{array}{ccc}1& 2& 2\\ 0& -2{k}^{2}-2k& -{k}^{2}+k+2\end{array}$

which is equivalent to

$\begin{array}{ccc}1& 2& 2\\ 0& 2k(k+1)& (k+1)(k-2)\end{array}$

From here we see that there is no solution iff $k=0$, an infinite number of solutions iff $k=-1$ and else there is exactly one solution.

Feinsn

Beginner2022-06-23Added 8 answers

Hint:

1. No solution when:

$\frac{1}{{k}^{2}}=\frac{2}{-2k}\ne \frac{1}{k+2}$

it holds when $k=0$

2. An infinite number of solutions when:

$\frac{1}{{k}^{2}}=\frac{2}{-2k}=\frac{1}{k+2}$

it holds when $k=-1$

3. Exactly one solution when:

$\frac{1}{{k}^{2}}\ne \frac{2}{-2k}$

and it hold when $k\ne -1$

1. No solution when:

$\frac{1}{{k}^{2}}=\frac{2}{-2k}\ne \frac{1}{k+2}$

it holds when $k=0$

2. An infinite number of solutions when:

$\frac{1}{{k}^{2}}=\frac{2}{-2k}=\frac{1}{k+2}$

it holds when $k=-1$

3. Exactly one solution when:

$\frac{1}{{k}^{2}}\ne \frac{2}{-2k}$

and it hold when $k\ne -1$

Find the volume V of the described solid S

A cap of a sphere with radius r and height h.

V=??

Whether each of these functions is a bijection from R to R.

a) $f(x)=-3x+4$

b) $f\left(x\right)=-3{x}^{2}+7$

c) $f(x)=\frac{x+1}{x+2}$

?

$d)f\left(x\right)={x}^{5}+1$In how many different orders can five runners finish a race if no ties are allowed???

State which of the following are linear functions?

a.$f(x)=3$

b.$g(x)=5-2x$

c.$h\left(x\right)=\frac{2}{x}+3$

d.$t(x)=5(x-2)$ Three ounces of cinnamon costs $2.40. If there are 16 ounces in 1 pound, how much does cinnamon cost per pound?

A square is also a

A)Rhombus;

B)Parallelogram;

C)Kite;

D)none of theseWhat is the order of the numbers from least to greatest.

$A=1.5\times {10}^{3}$,

$B=1.4\times {10}^{-1}$,

$C=2\times {10}^{3}$,

$D=1.4\times {10}^{-2}$Write the numerical value of $1.75\times {10}^{-3}$

Solve for y. 2y - 3 = 9

A)5;

B)4;

C)6;

D)3How to graph $y=\frac{1}{2}x-1$?

How to graph $y=2x+1$ using a table?

simplify $\sqrt{257}$

How to find the vertex of the parabola by completing the square ${x}^{2}-6x+8=y$?

There are 60 minutes in an hour. How many minutes are there in a day (24 hours)?

Write 18 thousand in scientific notation.