Systems of Linear Equations (DP IB Analysis & Approaches (AA): HL): Exam Questions

A festival charges $x$ dollars for an adult ticket, $y$ dollars for a child ticket and $z$ dollars for a car parking pass.

4 adult tickets, 7 child tickets and 2 car parking passes cost a total of $540. 2 adult tickets, 2 child tickets and 1 car parking pass cost a total of $210. 7 adult tickets and 3 car parking passes cost a total of $450.

(i) Write down a system of three linear equations in $x$, $y$ and $z$.

(ii) Hence find the values of $x$, $y$ and $z$.

Solve the following system of linear equations.  

$3x+2y-z=1$ 

$x-y+5z=-2$ 

$2x+y=3$

Solve the following system of linear equations.

$2x+2y-3z=-8$

$3x+2y-z=0$

$x-y+z=11$

Consider the system of equations 

$-6a+(k-3)b=1$ 

$3ka-5b=4$ 

Find the values of the real parameter $k$ such that the system has a unique solution.

Find the unique solution in terms of $k$.

Solve the following system of equations.

$3x+9y-3z=45$

$6x+3y+3z=21$

$3x-3y-6z=0$

Consider the following system of equations 

$2x+y-3z=-4$ 

$x-y+2z=2$ 

$4x+2y-6z=k$

where $k\in \mathrm{ℝ}$ 

Show that the system has no unique solution for any value of $k.$

Solve the following system of linear equations.

$2x-y+3z=4$

$3x+2y+6z=-5$

$2x-4y-z=8$

Solve the following system of linear equations.

$2x+3y+4z=15$

$x-2y-6z=-5$

$2x-6y-5z=6$

Solve the following system of linear equations.

$2x+2y-3z=8$

$3x-y+2z=6$

$x-2y+4z=-3$

Consider the polynomial $f\left(x\right)=x^4+a{x}^3+b{x}^2+cx+12$, where $a$, $b$ and $c$ are constants.

It is given that $f\left(1\right)=f\left(2\right)=f\left(3\right)=0$.

(i) Write down a system of three linear equations in $a$, $b$ and $c$.

(ii) Hence find the values of $a$, $b$ and $c$.

Solve the following system of linear equations.

$2x-3y-5z=4$

$x-4y+6z=-6$

$3y-2x-3z=0$

Solve the following system of linear equations.

$2x-3y+4z=-1$

$x-4y-6z=8$

$5y-3x-5z=1$

Solve the following system of linear equations.

$3y-2x+5z=14$

$3x-2y-2z=11$

$3z-4x-4y=35$

Find the unique point of intersection of the planes with the following Cartesian equations:

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

$z+2y-x=4$

$3z+y-4x=1$

Consider the function $f$ defined by $f\left(x\right)=x^4+a{x}^3+b{x}^2+cx+24$. The graph of $y=f\left(x\right)$ passes through the points $\left(1,36\right)$, $\left(2,24\right)$ and $\left(3,0\right)$.

Find the values of $a$, $b$ and $c$.

Consider the following system of linear equations.

 $2x+y-2z=6$

$2x-2y+3z=-5$

$-2x+3y+az=b$ 

Given that the system has no solutions, find the value of $a$ and the set of possible values of $b$.

Consider the following system of equations

$ax+y+z=4$

$x+y+z=a$

$x-y+az=2$

Find, in terms of $a$, expressions for $x,y$ and $z$.

Find the values of $a$ for which the system does not have a unique solution.

Given that $a=0$, find the values of $x$, $y$ and $z$.

The following system of equations has an infinite number of solutions.

$x-2y+z=k$

$x+y-z=2$

$3x-3y+z=12$

(i) Find the value of $k$.

(ii) Find a general solution.

Consider the system of equations

 $x+ky+z=k$

$x+2y+3z=0$

$3x+8y+5z=6$ 

When $k=m$, the system does not have a unique solution. Find the value of $m$.

Given that $k\ne m$, show that the solution to the system is independent of $k$ and hence find the unique solution.

Consider the system of equations

$2x+6y+pz=q$

$6y-x-z=7$

$x+2y-3z=1$

where $p$ and $q$ are real constants.

Find the value of $p$ such that the system does not have a unique solution.

Given that $z=0$, find the values of $x$, $y$ and $q$ for which the system has a solution.

Consider the following system of equations

$x+2y-z=0$

$2x-y+4z=0$

$x-3y+\left(1-a\right)z=4-a^2$

Show that the system has a unique solution when $a\ne -4$.

For the case where $a\ne -4$, find the solution in terms of $a$.

For the case where $a=-4$, show that there are no solutions.

The following system of equations has an infinite number of solutions.

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

$2x+y-4z=1$

$3y-x+3z=k$

Find the value of $k$.

Find the general solution.

Consider the following system of linear equations.

$x-3y+3z=k$

$3x-2y+z=4$

$2x+y-2z=2$

Show that the system of equations does not have a unique solution.

Find the value of $k$ for which the equations are consistent.

For the value of $k$ found in part (b), find the general solution of these equations.

The rate, $R$, at which the volume of a cloud created in a science lab increases is related to the air temperature, $T$, and the air pressure, $P$, by the equation

$R=k{T}^x{P}^y$

where $x$, $y$ and $k$ are real constants.

A scientist takes three measurements and records the results in the table.

Find $x$, $y$ and $k$.