Inbrunstlr

2022-10-06

We are required to solve the following system of equations:

$$\begin{array}{}\text{(1)}& {x}^{3}+\frac{1}{3{x}^{4}}=5\end{array}$$

$$\begin{array}{}\text{(2)}& {x}^{4}+\frac{1}{3{x}^{3}}=10\end{array}$$

We may multiply (1) by $3{x}^{4}$ throughout and (2) by $3{x}^{3}$ throughout (as 0 is not a solution we may cancel the denominators) to yield.

$$\begin{array}{}\text{(3)}& 3{x}^{7}+1=15{x}^{4}\end{array}$$

$$\begin{array}{}\text{(4)}& 3{x}^{7}+1=30{x}^{3}\end{array}$$

Subtracting the two:

$$15{x}^{4}-30{x}^{3}=0$$

$$\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}{x}^{3}(x-2)=$$

As $0$ is not a solution we choose $x=2$.

But putting $x=2$ in the original equations does not satisfy them. How come?

$$\begin{array}{}\text{(1)}& {x}^{3}+\frac{1}{3{x}^{4}}=5\end{array}$$

$$\begin{array}{}\text{(2)}& {x}^{4}+\frac{1}{3{x}^{3}}=10\end{array}$$

We may multiply (1) by $3{x}^{4}$ throughout and (2) by $3{x}^{3}$ throughout (as 0 is not a solution we may cancel the denominators) to yield.

$$\begin{array}{}\text{(3)}& 3{x}^{7}+1=15{x}^{4}\end{array}$$

$$\begin{array}{}\text{(4)}& 3{x}^{7}+1=30{x}^{3}\end{array}$$

Subtracting the two:

$$15{x}^{4}-30{x}^{3}=0$$

$$\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}{x}^{3}(x-2)=$$

As $0$ is not a solution we choose $x=2$.

But putting $x=2$ in the original equations does not satisfy them. How come?

odejicahfc

Beginner2022-10-07Added 10 answers

Solutions to (3) and (4) are equivalent to solutions of $15{x}^{4}-30{x}^{3}=0$ and one of (3) or (4). But the solution $15{x}^{4}-30{x}^{3}=0$ to $15{x}^{4}-30{x}^{3}=0$ does not satisfy (3) [or equivalently (4)], as $3\ast {2}^{7}+1\ne 15\ast {2}^{4}$.

You shouldn't be surprised that a system of two equations in one unknown may be inconsistent, i.e. not have any solutions.

You shouldn't be surprised that a system of two equations in one unknown may be inconsistent, i.e. not have any solutions.

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.