Systems of linear equations
In a farmyard there are cows and hens: 35 heads and 110 legs in total. A single equation is not enough for two unknowns, but here you have two clues: the number of heads and the number of legs. Putting together the two equations they give, the problem is solved. That is a system.
What a system of two equations is
To check, substitute into both: $20 + 15 = 35$ ✓ and $4 \cdot 20 + 2 \cdot 15 = 80 + 30 = 110$ ✓. If it satisfies only one, it is not a solution of the system.
Check whether the pair of values $x = 3$, $y = -2$ is a solution to the following system of equations. Justify your answer by showing your working. $$ \begin{cases} 4x +…
Check whether the pair of values $x = 3$, $y = -2$ is a solution to the following system of equations. Justify your answer by showing your working.
$$ \begin{cases} 4x + 3y = 6 \\ -2x + 5y = -16 \end{cases} $$On a farm there are cows and chickens. If in total there are $35$ heads and $110$ legs, set up a system of linear equations to find out how many animals of each kind…
On a farm there are cows and chickens. If in total there are $35$ heads and $110$ legs, set up a system of linear equations to find out how many animals of each kind there are.
Graphical meaning: two lines
Each equation of the system is a line. The solution of the system is the point where they intersect.
Look at the following graphical representation of a system of two linear equations with two unknowns. Determine the solution to the system from the graph and classify it…
Look at the following graphical representation of a system of two linear equations with two unknowns. Determine the solution to the system from the graph and classify it according to its number of solutions.
Classifying without solving: comparing coefficients
You can tell what type a system is without solving it, just by comparing the ratios between the coefficients of $\begin{cases} a_1x + b_1y = c_1 \\ a_2x + b_2y = c_2 \end{cases}$.
Classify the following systems as consistent with a unique solution, consistent with infinitely many solutions, or inconsistent, by analysing the ratio of their…
Classify the following systems as consistent with a unique solution, consistent with infinitely many solutions, or inconsistent, by analysing the ratio of their coefficients, without solving them. Justify your answer in each case.
- a) $ \begin{cases} 3x - 2y = 5 \\ -6x + 4y = -10 \end{cases} $
- b) $ \begin{cases} x + 4y = 7 \\ 2x + 8y = 12 \end{cases} $
- c) $ \begin{cases} 5x - y = 3 \\ 2x + 3y = 8 \end{cases} $
Classifying without dividing by zero
For two equations that represent lines, if $a_1b_2-a_2b_1\ne0$, there is a single solution. If it equals zero, compare the rows $(a_1,b_1,c_1)$ and $(a_2,b_2,c_2)$: proportional coefficients and constants in the same proportion give the same line; if the constants do not keep that proportion, there is no solution. An equation $0x+0y=0$ imposes no condition; $0x+0y=c$ with $c\ne0$ is impossible.
For example, $x=2$, $y=3$ has the unique solution $(2,3)$, even though the ratio method would divide by zero.
Systems with a parameter
When a system has a letter in addition to $x$ and $y$ (a parameter, usually $k$), you can use the same comparison of coefficients to study how the type of system changes according to its value.
Discuss the following system of equations in terms of the parameter $k$. Determine for which values of $k$ the system has no solution and for which it has a unique…
Discuss the following system of equations in terms of the parameter $k$. Determine for which values of $k$ the system has no solution and for which it has a unique solution.
$$ \begin{cases} kx + 2y = 6 \\ 3x + (k-1)y = 9 \end{cases} $$Going further: systems of three equations with three unknowns
The same idea extends to three unknowns with three equations. The usual tool is Gaussian elimination: use elimination to remove unknowns one by one until you are left with an equation in echelon form with a single unknown.
Once it is in echelon form, you solve from the bottom upwards: first $z$, then substitute to find $y$, and finally $x$.
Solve the following system of three equations with three unknowns using the elimination method (Gaussian elimination): $$ \begin{cases} x + 2y - z = 3 \\ 2x - y + 3z =…
Solve the following system of three equations with three unknowns using the elimination method (Gaussian elimination):
$$ \begin{cases} x + 2y - z = 3 \\ 2x - y + 3z = 10 \\ -x + 3y + 2z = 5 \end{cases} $$