Answer to Question #295394 in Algebra for Math assignment

Question

21. Find the values of my for which the system is not a Cramer's system.

(a)-2x+m²y=m

x-my=1-m

(b) mx+y=2

x+my=-2

22.(a) Mr. Benny is now 3 times as old as his son. In 13 years' time he will be twice as old as his son,Find Mr. Benny's present age.

(b) Two buses are 200 kilometers apart. They are traveling towards each other, the first bus is traveling 40km per hour while the second bus is traveling 60km per hour. When and where will they meet?

23.Two consecutive intergers are such that their reciprocatals add to 9/10. Find the two integers.

Answer 1

21.

a)

$$\Delta=\begin{vmatrix} -2 & m^2 \\ 1 & -m \end{vmatrix}=2m-m^2=0$$

$$m_1=0, m_2=2$$

The system is not a Cramer's system for $m=0$ or $m=2.$

b)

$$\Delta=\begin{vmatrix} m & 1\\ 1 & m \end{vmatrix}=m^2-1=0$$

$$m_1=-1, m_2=1$$

The system is not a Cramer's system for $m=-1$ or $m=1.$

22.

a) Let $x=$ sun's present age

Then

$$3x+13=2(x+13)$$

$$x=13$$

$$3(13)=39$$

Mr. Benny is 39 years old now.

b)

$$t=\dfrac{200\ km}{40\ km/h+60\ km/h}=2\ h$$

The buses will meet 2 hours after the start.

First bus will travell $40\ km/h(2\ h)=80\ km.$

Second bus will travell $60\ km/h(2\ h)=120\ km.$

23.

$$\dfrac{1}{n}+\dfrac{1}{n+1}=\dfrac{9}{10}, n\in \Z$$

$$10n+10n+10=9n^2+9n$$

$$9n^2-11n-10=0$$

$$D=(-11)^2-4(9)(-10)=481$$

Hence the eqution $9n^2-11n-10=0$ has no integer solution.

$$\dfrac{1}{n}+\dfrac{1}{n+1}=\dfrac{9}{20}, n\in \Z$$

$$20n+20n+20=9n^2+9n$$

$$9n^2-31n-20=0$$

$$D=(-31)^2-4(9)(-20)=1681=41^2$$

$$n=\dfrac{31\pm41}{2(9)}$$

$$n_1=-\dfrac{5}{9}, n_2=4$$

Since $n\in \Z,$ we take $n=4.$

Answer: 4 and 5.