Linear Equations

If x⁴ - x³ + ax² + x + b is exactly divisible by x² - 3x + 2, then the value of a and b are respectively.

Using matrices, solve the following system of equations

 $\begin{array}{l}4x-5y-11z=12\end{array}$

$x-3y+z=1$

$2x+3y-7z=2$

Using matrices, what would be the value of $x, y, z$ for the following system of equations :

$\begin{array}{l}x+2y+z=7\end{array}$

$x+3z=11$

$2x-3y=1$

If a point $P\left(\alpha ,\beta ,\gamma \right)$ satisfying $\left(\begin{array}{ccc}\alpha & \beta & \gamma \end{array}\right)\left(\begin{array}{ccc}2& 10& 8\\ 9& 3& 8\\ 8& 4& 8\end{array}\right)=\left(\begin{array}{lll}0& 0& 0\end{array}\right)$ lies on the plane $2x+4y+3z=5$, then $6\alpha +9\beta +7\gamma$ is equal to

Which equation has a solution $-2$?

If the lines $$\begin{gathered} \lambda x+\left(\sin \alpha \right)y+\cos \alpha =0, \\ x+\left(\cos \alpha \right)y+\sin \alpha =0, \\ x-\left(\sin \alpha \right)y+\cos \alpha =0 \end{gathered}$$ pass through the same point where $\alpha \in R$ and $\lambda$ lies in the interval:

Let $x=\alpha ,y=\beta ,z=\gamma$ be the unique solution of the system of simultaneous linear equations $2x+3y-2z+4=0,3x-4y+3z+5=0,kx-2y+z+3=0$. If $\alpha =-2$ then $k=$

If $\frac{{a_1}^2+{a_2}^2}{{a_1}^2-{a_2}^2}=\frac{5}{3}$, then find $\frac{a_2}{a_1}$.

How many values for $k$ the system of equations $(k+1)x+8y=4k \text{and} kx+(k+3)y=3k-1$ have an infinitely many solution?

What should be the value of $k$ so that the system of linear equations $x-y+2z=0,kx-y+z=0,3x+y-3z=0$ does not possess a unique solution?

An airplane travels $910$ miles with a tailwind in the same time that it travels $660$ miles with headwind. The speed of the airplane is $305$ miles per hour in still air. Determine the wind's speed.

Solve using equivalence transformations:

$\frac{3x}{2}=10$

Solve using equivalence transformations:

$4(x-1)=2x+7$

Solve using equivalence transformations:

$2x+3=5x-9$

Solve the given system of equations using the inverse of a matrix.

$$\begin{gathered} x+2y=4 \\ 3x-5y=1 \end{gathered}$$