System of Linear Equations Archives

System of Linear Equations

Two schools A and B want to award their selected students on the values of sincerity, truthfulness and helpfulness. The school A wants to award ₹ x each, ₹ y each and ₹ z each for the three respective values to 3, 2 and 1 students respectively with total award money of ₹ 1,600. School B wants to spend ₹ 2,300 to award its 4, 1 and 3 students on the respective values (by giving the same award money to the three values as before). If the total amount of award for one prize on each value is ₹ 900, using matrices, find the award money for each value. Apart from these three values, suggest one more value which should be considered for award. HINT: By the given data, we have
$3 x+2 y+z=1600$
$4x+y+3 z=2300$
$x+y+z=900$

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: Assume the amount considered for sincerity, truthfulness and helpfulness are $x, y$ and $z$ respectively. As per the questions, $3 x+2 y+z=1600$ $\begin{array}{l} 4 x+y+3 z=2300 \\...

An amount of ₹ 5000 is put into three investments at 6%, 7% and 8% per annum respectively. The total annual income from these investments is ₹358. If the total annual income from first two investments is ₹70 more than the income from the third, find the amount of each investment by the matrix method. HINT: Let these investments be ₹x, ₹y and ₹z, respectively. Then, $x+y+z=5000, \ldots$ (i) $\begin{array}{l} \frac{6 x}{100}+\frac{7 y}{100}+\frac{8 z}{100}=358 \Rightarrow \\ 6 x+7 y+8 z=35800 \ldots (ii) \end{array}$ And, $\frac{6 x}{100}+\frac{7 y}{100}=\frac{8 z}{100}+70$ $\Rightarrow 6 x+7 y-8 z=7000 . \ldots \text { (iii) }$

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: Suppose the investments are $\mathrm{x} \mathrm{x}$, Fy and $\mathrm{F} \mathrm{z}$, respectively. Therefore, $x+y+z=5000$ $\begin{array}{l} \frac{6 x}{100}+\frac{7 y}{100}+\frac{8...

The cost of 4 kg potato, 3 kg wheat and 2 kg of rice is ₹ 60. The cost of 1 kg potato, 2 kg wheat and 3 kg of rice is ₹45. The cost of 6 kg potato, 2 kg wheat and 3 kg of rice is ₹70. Find the cost of each item per kg by matrix method.

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: Suppose the price of 1kg potato, wheat and rice is $x$, $y$ and $z$ respectively. As per the question, $4x + 3y + 2z = 60$ $x+ 2y + 3z = 45$ $6x + 2y + 3z = 70$ Now converting the...

The sum of three numbers is 2. If twice the second number is added to the sum of first and third, we get 1. On adding the sum of second and third numbers to five times the first, we get 6. Find the three numbers by using matrices.

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: Assume the numbers are $\mathrm{x}, \mathrm{y}$ and $\mathrm{z}$. As per the question, $\begin{array}{l} x+y+z=2 \\ x+2 y+z=1 \\ 5 x+y+z=6 \end{array}$ Now converting the following...

$\begin{array}{l} \frac{1}{x}-\frac{1}{y}+\frac{1}{z}=4 ; \frac{2}{x}+\frac{1}{y}-\frac{3}{z}=0 \\ \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=2 \cdot(x, y, z \neq 0) \end{array}$

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: We need to find: $-x, y, z$, The given set of lines are : $\begin{array}{l} \frac{1}{x}-\frac{1}{y}+\frac{1}{z}=4 \\ \frac{2}{x}+\frac{1}{y}-\frac{3}{z}=0 \\...

$\begin{array}{l} \frac{2}{x}-\frac{3}{y}+\frac{3}{z}=10, \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=10 \\ \frac{3}{x}-\frac{1}{y}+\frac{2}{z}=13 \end{array}$

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: We need to find: $-x, y, z$ The given set of lines are : - $\begin{array}{l} \frac{2}{x}-\frac{3}{y}+\frac{3}{z}=10 \\ \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=10 \\...

If $A=\left(\begin{array}{ccc}1 & -2 & 0 \\ 2 & 1 & 3 \\ 0 & -2 & 1\end{array}\right)$ and $B=\left(\begin{array}{ccc}7 & 2 & -6 \\ -2 & 1 & -3 \\ -4 & 2 & 5\end{array}\right)$ , find $A B$ Hence, solve the system of equations:
$x-2 y=10$
$2 x+y+3 z=8$ and
$-2 y+z=7$ HINT: $A B=(11) /=A\left(\frac{1}{11} B\right)=/$ $A^{-1}=\left(\frac{1}{11}\right) B$

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: It is given, $\begin{array}{l} A=\left[\begin{array}{ccc} 1 & -2 & 0 \\ 2 & 1 & 3 \\ 0 & -2 & 1 \end{array}\right], B=\left[\begin{array}{ccc} 7 & 2 & -6 \\...

If $A=\left(\begin{array}{ccc}2 & 1 & 1 \\ 1 & -2 & -1 \\ 0 & 3 & -5\end{array}\right)$ , find $A^{-1}$ Using $A^{-1}$ , solve the following system of linear equations:
$2 x+y+z=1$
$x-2 y-z=\frac{3}{2}$
$3 y-5 z=9$ HINT: Here $A=\left(\begin{array}{ccc}2 & 1 & 1 \\ 1 & -2 & -1 \\ 0 & 3 & -5\end{array}\right)$ $\mathrm{X}=\left(\begin{array}{l} x \\ y \\ z \end{array}\right) \text { and } \mathrm{B}=\left(\begin{array}{l} 1 \\ 3 / 2 \\ 9 \end{array}\right)$

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: It is given, $\begin{array}{l} A=\left[\begin{array}{ccc} 2 & 1 & 1 \\ 1 & -2 & -1 \\ 0 & 3 & -5 \end{array}\right] \\ A^{-1}=\frac{1}{|A|} \operatorname{adj}(A)...

If $A=\left(\begin{array}{ccc}2 & -3 & 5 \\ 3 & 2 & -4 \\ 1 & 1 & -2\end{array}\right)$ , find $A^{-1}$ . Using $A^{-1}$ , solve the following system of equations:
$2 x-3 y+5 z=11$
$3 x+2 y-4 z=-5$
$x+y-2 z=-3$

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: It is given, $A=\left[\begin{array}{ccc} 2 & -3 & 5 \\ 3 & 2 & -4 \\ 1 & 1 & -2 \end{array}\right]$ $\mathrm{A}^{-1}=\frac{1}{|A|} \operatorname{adj}(A)$...

Solve each of the following systems of equations using matrix method.
$4x + 3y + 2z = 60$ ;
$x + 2y + 3z = 45$ ;
$6x + 2y + 3z = 70$ .

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: We need to find: - $x , y , z$ The given set of lines are : $\begin{array}{l} 4 x+3 y+2 z=60 \\ x+2 y+3 z=45 \\ 6 x+2 y+3 z=70 \end{array}$ Now converting the following equations in matrix...

Solve each of the following systems of equations using matrix method.
$x - y = 3$ ;
$2x + 3y + 4z = 17$ ;
$y + 2z = 7$ .

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: We need to find: $-\mathrm{x}, \mathrm{y}, \mathrm{z}$ The given set of lines are : - $\begin{array}{l} x-y=3 \\ 2 x+3 y+4 z=17 \\ y+2 z=7 \end{array}$ Now, converting the following...

Solve each of the following systems of equations using matrix method.
$x - 2y + z = 0$ ;
$y - z = 2$ ;
$2x - 3z = 10$ .

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: We need to find: $-\mathrm{x}, \mathrm{y}, \mathrm{z}$ The given set of lines are : - $\begin{array}{l} x-2 y+z=0 \\ y-z=2 \\ 2 x-3 z=10 \end{array}$ Now converting the following equations...

Solve each of the following systems of equations using matrix method.
$5x - y = - 7$ ;
$2x + 3z = 1$ ;
$3y - z = 5$ .

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: We need to find: $-\mathrm{x}, \mathrm{y}, \mathrm{z}$ The given set of lines are : - $\begin{array}{l} 5 x-y=-7 \\ 2 x+3 z=1 \\ 3 y-z=5 \end{array}$ Now, converting the following...

Solve each of the following systems of equations using matrix method.
$x - y - 2z = 3$ ;
$x + y = 1$ ;
$x + z = - 6$ .

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: We need to find: $-\mathrm{x}, \mathrm{y}, \mathrm{z}$ The given set of lines are : - $\begin{array}{l} x-y-2 z=3 \\ x+y=1 \\ x+z=-6 \end{array}$ Now converting the following equations in...

Solve each of the following systems of equations using matrix method.
$x + 2y + z = 4$ ;
$- x + y + z = 0$ ;
$x - 3y + z = 4$ .

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: We need to find: $-x, y, z$ The given set of lines are : - $\begin{array}{l} x+2 y+z=4 \\ -x+y+z=0 \\ x-3 y+z=4 \end{array}$ Now converting the following equations in matrix form,...

Solve each of the following systems of equations using matrix method.
$2x + y - z = 1$ ;
$x - y + z = 2$ ;
$3x + y - 2z = - 1$ .

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: We need to find: $-\mathrm{x}, \mathrm{y}, \mathrm{z}$ The given set of lines are : - $\begin{array}{l} 2 x+y-z=1 \\ x-y+z=2 \\ 3 x+y-2 z=-1 \end{array}$ Now, converting the following...

Solve each of the following systems of equations using matrix method.
$x + y - z = 1$ ;
$3x + y - 2z = 3$ ;
$x - y - z = - 1$ .

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: We need to find: - $x , y , z$ The given set of lines are : - $x + y - z = 1$ $\begin{array}{l} 3 x+y-2 z=3 \\ x-y-z=-1 \end{array}$ Now, converting the following equations in matrix form,...

Solve each of the following systems of equations using matrix method.
$3x - 4y + 2z = - 1$ ;
$2x + 3y + 5z = 7$ ;
$x+ z = 2$ .

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: We need to find: - $x , y , z$ The given set of lines are : - $\begin{array}{l} 3 x-4 y+2 z=-1 \\ 2 x+3 y+5 z=7 \\ x+z=2 \end{array}$ Now, converting the following equations in matrix...

Solve each of the following systems of equations using matrix method.
$6x - 9y - 20z = - 4$ ;
$4x - 15y + 10z = - 1$ ;
$2x - 3y - 5z = - 1$ .

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: We need to find: $-\mathrm{x}, \mathrm{y}, z$ The given set of lines are : - $\begin{array}{l} 6 x-9 y-20 z=-4 \\ 4 x-15 y+10 z=-1 \\ 2 x-3 y-5 z=-1 \end{array}$ Now, converting the...

Solve each of the following systems of equations using matrix method.
$x - y + 2z = 7$ ;
$3x + 4y - 5z = - 5$ :
$2x - y + 3z = 12$ .

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: We need to find: $-\mathrm{x}, \mathrm{y}, \mathrm{z}$ The given set of lines are : - $\begin{array}{l} x-y+2 z=7 \\ 3 x+4 y-5 z=-5 \\ 2 x-y+3 z=12 \end{array}$ Now, converting the...

Solve each of the following systems of equations using matrix method.
$4x - 5y - 11z = 12$ ;
$x - 3y + z = 1$ ;
$2x + 3y - 7z = 2$ .

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: We need to find: $-\mathrm{x}, \mathrm{y}, \mathrm{z}$ The given set of lines are: $\begin{array}{l} 4 x-5 y-11 z=12 \\ x-3 y+z=1 \\ 2 x+3 y-7 z=2 \end{array}$ Now, converting the...

Solve each of the following systems of equations using matrix method.
$2x + 3y + 3z = 5$ ;
$x - 2y + z = - 4$ ;
$3x - y - 2z = 3$ .

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: We need to find: - $x, y, z$ The given set of lines are : - $\begin{array}{l} 2 x+3 y+3 z=5 \\ x-2 y+z=-4 \\ 3 x-y-2 z=3 \end{array}$ Now, converting the following equations in matrix...

Solve each of the following systems of equations using matrix method.
$x + y + z = 6$ ;
$x + 2z = 7$ ;
$3x + y + z = 12$ .

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: We need to find: $-\mathrm{x}, \mathrm{y}, \mathrm{z}$ The given set of lines are : - $x+y+z=6$ $\begin{array}{l} x+2 z=7 \\ 3 x+y+z=12 \end{array}$ Now converting following equations in...

Solve each of the following systems of equations using matrix method.
$x + y + z = 1$ ;
$x - 2y + 3z = 2$ ;
$5x - 3y + z = 3$ .

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: We need to find: $-\mathrm{x}, \mathrm{y}, \mathrm{z}$ The given set of lines are : - $\begin{array}{l} x+y+z=1 \\ x-2 y+3 z=2 \\ 5 x-3 y+z=3 \end{array}$ Now converting the following...

Solve each of the following systems of equations using matrix method.
$2x - 3y + 5z = 11$ ;
$3x + 2y - 4z = - 5$ ;
$x + y - 2z = - 3$ .

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: We need to find: $-\mathrm{x}, \mathrm{y}, \mathrm{z}$ The given set of lines are : - $\begin{array}{l} 2 x-3 y+5 z=11 \\ 3 x+2 y-4 z=-5 \\ x+y-2 z=-3 \end{array}$ Now converting the...

Solve each of the following systems of equations using matrix method.
$x + y + z = 4$ ;
$2x - y + z = - 1$ ;
$2x + y - 3z = - 9$ .

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: We need to find: $-\mathrm{x}, \mathrm{y}, \mathrm{z}$ The given set of lines are : - $\begin{array}{l} x+y+z=4 \\ 2 x-y+z=-1 \\ 2 x+y-3 z=-9 \end{array}$ Now, converting the following...

Solve each of the following systems of equations using matrix method.
$2x - 3y + 5z = 16$ ;
$3x + 2y - 4z = - 4$
$x + y - 2z = - 3$ .

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: We need to find: $-\mathrm{x}, \mathrm{y}, \mathrm{z}$ The given set of lines are : - $\begin{array}{l} 2 x-3 y+5 z=16 \\ 3 x+2 y-4 z=-4 \\ x+y-2 z=-3 \end{array}$ Now converting the...

Solve each of the following systems of equations using matrix method.
$x + 2y + z = 7$ ;
$x + 3z = 11$ ;
$2x - 3y = 1$ .

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: We need to find: $-\mathrm{x}, \mathrm{y}, \mathrm{z}$ The given set of lines are : - $\begin{array}{l} x+2 y+z=7 \\ x+3 z=11 \\ 2 x-3 y=1 \end{array}$ Now converting following equations...

Solve each of the following systems of equations using matrix method.
$3x + 4y + 7z = 4$ ;
$2x - y + 3z = - 3$ ;
$x + 2y - 3z = 8$ .

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: We need to find: $-\mathrm{x}, \mathrm{y}, \mathrm{z}$ The given set of lines are : - $\begin{array}{l} 3 x+4 y+7 z=4 \\ 2 x-y+3 z=-3 \\ x+2 y-3 z=8 \end{array}$ Now converting the...

Solve each of the following systems of equations using matrix method.
$x - y + z = 1$ ;
$2x + y - z = 2$ ;
$x - 2y - z = 4$ .

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: We need to find: $-x, y, z$ The given set of lines are : - $\begin{array}{l} x-y+z=1 \\ 2 x+y-z=2 \\ x-2 y-z=4 \end{array}$ Now converting following equations in matrix form,...

Solve each of the following systems of equations using matrix method.
$2x + 8y + 5z = 5$ ;
$x + y + z = - 2$ ;
$x + 2y - z = 2$ .

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: We need to find: $-\mathrm{x}, \mathrm{y}, \mathrm{z}$ The given set of lines are: $\begin{array}{l} 2 x+8 y+5 z=5 \\ x+y+z=-2 \\ x+2 y-z=2 \end{array}$ Now, converting the following...

Solve each of the following systems of equations using matrix method.
$4x - 3y = 3$ ;
$3x - 5y = 7$ .

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: We need to find: $-\mathrm{x}, \mathrm{y}$ The given set of lines are : - $\begin{array}{l} 4 x-3 y=3 \\ 3 x-5 y=7 \end{array}$ Now, converting the following equations in matrix form,...

Solve each of the following systems of equations using matrix method.
$2x - 3y + 1 = 0$ ;
$x + 4y + 3 = 0$ .

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: We need to find: - $x, y$ The given set of lines are : - $\begin{array}{l} 2 \mathrm{x}-3 \mathrm{y}+1=0 \\ \mathrm{x}+4 y+3=0 \end{array}$ On converting the following equations in matrix...

Solve each of the following systems of equations using matrix method.
$5x + 7y + 2 = 0$ ;
$4x + 6y + 3 = 0$ .

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: We need to find: $-\mathrm{x}, \mathrm{y}$ The given set of lines are : - $\begin{array}{l} 5 \mathrm{x}+7 \mathrm{y}+2=0 \\ 4 \mathrm{x}+6 \mathrm{y}+3=0 \end{array}$ On converting the...

Solve each of the following systems of equations using matrix method.
$x + 2y = 1$ ;
$3x + y = 4$ .

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: We need to find: $-\mathrm{x}, \mathrm{y}$ The given set of lines are : - $\begin{array}{l} x+2 y=1 \\ 3 x+y=4 \end{array}$ Now converting the following equations in matrix form,...

Solve each of the following systems of equations using matrix method.
$3x + 4y - 5 = 0$ ;
$x - y + 3 = 0$ .

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: We need to find: $-\mathrm{x}, \mathrm{y}$ The given set of lines are : - $\begin{array}{l} 3 x+4 y-5=0 \\ x-y+3=0 \end{array}$ On converting the following equations in matrix form,...

Solve each of the following systems of equations using matrix method.
$5x + 2y = 4$ ;
$7x + 3y = 5$ .

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: We need to find: $-\mathrm{x}, \mathrm{y}$ The given set of lines are : - $\begin{array}{l} 5 x+2 y=4 \\ 7 x+3 y=5 \end{array}$ On converting the following equations in matrix form,...

Show that each one of the following systems of equations is inconsistent.
$3x - y - 2z = 2$ ;
$2y - z = - 1$ ;
$3x - 5y = 3$ .

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: We need to prove: Set of given lines are inconsistent. The given set of lines are : - $\begin{array}{l} 3 x-y-2 z=2 \\ 2 y-z=-1 \\ 3 x-5 y=3 \end{array}$ Now, converting the following...

Show that each one of the following systems of equations is inconsistent.
$x + 2y + 4z = 12$ ;
$y + 2z = - 1$ ;
$3x + 2y + 4z = 4$ .

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: We need to prove: let of given lines are inconsistent. The given set of lines are : - $\begin{array}{l} x+2 y+4 z=12 \\ y+2 z=-1 \\ 3 x+2 y+4 z=4 \end{array}$ Now, converting the following...

Show that each one of the following systems of equations is inconsistent.
$2x - y + 3z = 1$ ;
$3x - 2y + 5z = - 4$ ;
$5x - 4y + 9z = 14$ .

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: We need to prove: Set of given lines are inconsistent. The given set of lines are : - $\begin{array}{l} 2 x-y+3 z=1 \\ 3 x-2 y+5 z=-4 \\ 5 x-4 y+9 z=14 \end{array}$ On converting the...

Show that each one of the following systems of equations is inconsistent.
$x + y - 2z = 5$ ;
$x - 2y + z = - 2$ ;
$-2x + y + z = 4$ .

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: We need to prove: Set of given lines are inconsistent. The given set of lines are : - $\begin{array}{l} x+y-2 z=5 \\ x-2 y+z=-2 \\ -2 x+y+z=4 \end{array}$ Now, converting the following...