Solving a system of linear equation in two variables

Q. (i) If $A=\left[\begin{array}{ccc}1& -2& 0\\ 2& 1& 3\\ 0& -2& 1\end{array}\right]$, find A⁻¹. Using A⁻¹, solve the system of linear equations
x − 2y = 10, 2x + y + 3z = 8, −2y + z = 7

(ii) $A=\left[\begin{array}{ccc}3& -4& 2\\ 2& 3& 5\\ 1& 0& 1\end{array}\right]$, find A⁻¹ and hence solve the following system of equations:
3x − 4y + 2z = −1, 2x + 3y + 5z = 7, x + z = 2

(iii) $A=\left[\begin{array}{ccc}1& -2& 0\\ 2& 1& 3\\ 0& -2& 1\end{array}\right]\mathrm{and}B=\left[\begin{array}{ccc}7& 2& -6\\ -2& 1& -3\\ -4& 2& 5\end{array}\right]$, find AB. Hence, solve the system of equations:
x − 2y = 10, 2x + y + 3z = 8 and −2y + z = 7

(iv) If $A=\left[\begin{array}{ccc}1& 2& 0\\ -2& -1& -2\\ 0& -1& 1\end{array}\right]$, find A⁻¹. Using A⁻¹, solve the system of linear equations
x − 2y = 10, 2x − y − z = 8, −2y + z = 7

(v) Given $A=\left[\begin{array}{ccc}2& 2& -4\\ -4& 2& -4\\ 2& -1& 5\end{array}\right],B=\left[\begin{array}{ccc}1& -1& 0\\ 2& 3& 4\\ 0& 1& 2\end{array}\right]$, find BA and use this to solve the system of equations
y + 2z = 7, x − y = 3, 2x + 3y + 4z = 17

Q. An equation that can be used to convert a temperature, in degrees Fahrenheit ${(}^{\circ }F)$, to a temperature in degrees in Celsius ${(}^{\circ }C)$ is $c=\frac{5}{9}(f-32)$ . For what value is the temperature in ${(}^{\circ }F)$ equal to the temperature in ${(}^{\circ }C)$?

1. -40
2. 40
3. -100
4. -32

Q.

How do you solve a word problem with $3$ variables?

Q. $-33y=6(3z+3)-39y$, $6y-22=c(z-3)$
For what value of $c$ does the above system of linear equations in the variables $y$ and $z$ have infinitely many solutions?

1. 5/7
2. 17/15
3. 7/11
4. 11/2

Q. $y=\frac{4}{7}x$
$\frac{2}{3}x=y+\frac{5}{7}$

Consider the system of equations above. If $(x,y)$ is the solution to the system, then what is the value of $y$?

1. $\frac{30}{7}$
2. $\frac{15}{2}$
3. $\frac{2}{21}$
4. $Noneoftheabove$

Q.

Five years ago, A was three times as old as B and ten years later A shall be twice as old as B. What are the present ages of A and B (in years)?

1. $45,15$

2. $30,40$

3. $50,30$

4. $50,20$

Q. 7x + 3 y = 15;
14x + 24y = 10
Which ordered pair (x, y) satisfies the system of equations above?

1. $(\frac{-55}{21},\frac{-10}{9})$
2. $(\frac{40}{21},\frac{5}{9})$
3. $(\frac{55}{21},\frac{-10}{9})$
4. $(\frac{40}{21},\frac{-5}{9})$

Q. Using elimination method, solve $101x+99y=499,9x+101y=501$.

Q. px + 3y = 2(1 - y) + 1
5 + y = 3(1 + y) + 2x
In the system of linear equations above, p is a constant. For what value of p does the equation have exactly one solution (x, y) with y = 2?

1. -1/7
2. 7
3. -11
4. 11/7

Q. Show that each one of the following systems of linear equation is inconsistent:
(i) 2x + 5y = 7
6x + 15y = 13

(ii) 2x + 3y = 5
6x + 9y = 10

(iii) 4x − 2y = 3
6x − 3y = 5

(iv) 4x − 5y − 2z = 2
5x − 4y + 2z = −2
2x + 2y + 8z = −1

(v) 3x − y − 2z = 2
2y − z = −1
3x − 5y = 3

(vi) x + y − 2z = 5
x − 2y + z = −2
−2x + y + z = 4

Q.

Solve the following systems of equations:
$\frac{xy}{x+y}=\frac{6}{5}$

$\frac{xy}{(y-x)}=6,$
where $x+y\ne 0$ and $y-x\ne 0$

Q.

For what value of $c$, the linear equation $2x+cy=8$ has equal values of $x$ and $y$ for its solution.

Q. $\frac{1}{2}x-2y=5$;
$-\frac{1}{2}x+3y=4$
If (x , y) are defined as the solution of the system of equations shown above, then what is the value of x if y= 9 is a solution. ?

1. x =46
2. x=23
3. x= 30
4. x = 33

Q. $2a(2p-q)=1$,
$p=2q-1$
Consider the system of equations above, where $a$ is a constant. For which value of $a$ is $(p,q)=(1,1)$ a solution?

1. None of the above

2. $2$

3. $\frac{1}{2}$
4. $-\frac{1}{2}$

Q.

State which of the following are equations (with a variable). Give a reason for your answer. Identify the variable from the equations with a variable.

$5\times 4-8=2x$

Q. $6000=bu+2000$
Given the above equation with constant $bu$, what is the value of $600,000+100bu+300,000$ in millions?

1. 1.2
2. 1.4
3. 1.7
4. 1.3

Q.

Solution of the inequality |3 - ${\log }_{2}x$| < 2 contains the interval.

1. x ∈ (2, 32)

2. x > 2

3. x ∈ [2, 32]

4. x < 32

Q. 2(2y + 3) = 3x - 5 ;
-2x = 4y + 6
Consider the above system of equations. Which of the following solutions satisfy the sytem ?

1. $(2,-2)$
2. $(1,-2)$
3. $(0,-3)$
4. $(\frac{1}{2},2)$

Q. The existence of the unique solution of the system of equations:
x + y + z = λ
5x − y + µz = 10
2x + 3y − z = 6
depends on
(a) µ only
(b) λ only
(c) λ and µ both
(d) neither λ nor µ

Q. $-4x-6-2y=-4y+8y+2$ ,
$2x+4=12$
Consider the system of equations above. If $(x,y)$ is the solution to the system, then what is the value of $x+y$ ?

1. $-8$
2. 0
3. 8
4. There is no solution to this system of equations

Q. If $a^b=4-ab$ and $b^a=1$, where $a$ and $b$ are positive integers, find $a$.

1. $0$
2. $1$
3. $2$
4. $3$

Q. If two real numbers x and y satisfy the equation x/y = x- y, then:

1. x$\ge$4 and x$\le$0 where x$\ge$ a means that x can take any value greater than a or equal to a
2. y= can equal 1
3. both x and y must be irrational
4. x and y cannot both be integers
5. both x and y must be rational

Q.

If $\left(107\right)\times 8=\left(y\right)\times 2$, find the value of $y$.

Q. Let a, b, c be positive real numbers. The following system of equations in x, y and z
$\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1,\frac{x^2}{a^2}-\frac{y^2}{b^2}+\frac{z^2}{c^2}=1,-\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1\mathrm{has}$

(a) no solution
(b) unique solution
(c) infinitely many solutions
(d) finitely many solutions

Q. $3x+4y=0$ , $\frac{2}{3}y-\frac{3}{4}x=20$
Consider the system of equations above. If (x, y) is the solution to the system, then what is the value of $\frac{x}{y}$?

1. $-\frac{4}{3}$
2. $-\frac{3}{4}$
3. $\frac{3}{4}$
4. $\frac{4}{3}$

Q. $4a-6b=9$ , $b+4=8a$
Consider the system of equations above. Which of the following statement is true?

1. There is only one solution $(a,b)\&a+b$ is negative
2. There are no solutions
3. There is only one solution $(a,b)\&a+b$ is positive
4. There are infinitely many solutions

Q.

Examine the consistency of the system of equations

$2x-y=5,x+y=4$

Q. $8p-4t=12$
$24p+18t=36+6t$
Which of the following accurately describes all solutions to the system of equations shown above?

1. $p=\frac{3}{2}\&t=0$
2. $p=3\&t=3$
3. $Therearenosolutiontothesystem.$
4. $Thereareinfinitesolutionstothesystem.$

Q. x + y = 7;
3x + 4y = 11
What value of (x , y) satisfies the above system of equations?

1. (17, -10)
2. (-17, 10)
3. (-10, 17)
4. (10, -17)

Q. Equations having a common solution are called

1. Linear equtions
2. Homogeneous equations
3. Simultaneous equations
4. None of the Above