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

Solve the following simultaneous equations. 

$\frac{1}{2}x+y=9$

$x-2y=2$

$3a-5b=30$

$5a+2b=3.5$

$\frac{5m}{6}-\frac{3n}{4}=2n$

$\frac{2m}{3}-\frac{4n}{5}=-\frac{2}{3}$

Use an algebraic method to solve the following system of linear equations.

$2x-y+3z=4$

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

$2x-4y-z=8$

Use an algebraic method to solve the following system of linear equations.

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

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

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

Use an algebraic method to solve the following system of linear equations.

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

$3x-y+2z=6$

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

Two straight lines have equations $y=\frac{3}{2}x+1$ and $4x-3y-1=0$. Find the coordinates of their point of intersection.

Consider the polynomial $f\left(x\right)=x^4+a{x}^3+b{x}^2+cx+12$ 

Given that $f\left(1\right)=f\left(2\right)=f\left(3\right)=0,$ 

(i) set up a system of linear equations in three unknowns,

(ii) hence, 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 value(s) of the real parameter $a$ so that the system has no unique solutions.

Given $a=0$, 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 the 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.

Solve the following simultaneous equations.

$5x-2y=9.5$

$-2x-5y=16.5$

$3\left(a-2b\right)=7-b$

$2\left(2a-b\right)=5b-11$

$\frac{2\left(p-q\right)}{5}-\frac{3p}{2}=\frac{p-4q}{10}$

$\frac{2p}{3}-\frac{4q}{5}=1-\frac{2}{3}q$

Use an algebraic method to 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 using an algebraic method.

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

$x-4y-6z=8$

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

Use an algebraic method to 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 $y=f\left(x\right)$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 system of equations

$2a+6b+xc=y$

$6b-a-c=7$

$a+2b-3c=1$

where $x$ and $y$ are real constants.

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

Given that $c=0$ find the values of $a,b$ and $y$ such that the system of equations 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 $a\ne -4$, state 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, $R$, of increase of the volume of a cloud created in a science lab is related to the change in air temperature,$T$ , and air pressure, $P$, by the equation

 $R=k{T}^x{P}^y$, where $x,y,k\in \mathrm{ℝ}$.

A meteorologist takes measurements at three intervals and records the data as follows.

Find $x,y$and $k$.