Unit 1 - Number and Algebra
Get Ready
Questions
Solve this pair of simultaneous equations using different methods$$\begin{matrix}2x&+&5y&=&16\\5x&-&2y&=&11\end{matrix}$$
Solutions
Notes
When solving a system of two linear equations there are three possible outcomes:
There is a unique solution - the lines intersect once
There are no solutions - the lines are parallel
There are infinitely many solutions - the lines are the same (perhaps written differently)
There are three method we can use to solve a system of two linear equations:
Elimination
Substitution
Matrix Row Reduction
In an exam paper where calculators are allowed, you can use your GDC to solve a system of equations.
Examples and Your Turns
Example
Solve this pair of simultaneous equations using matrix row reduction$$\begin{matrix}2x&+&5y&=&16\\5x&-&2y&=&11\end{matrix}$$
Your Turn
Solve this pair of simultaneous equations using matrix row reduction$$\begin{matrix}2x&+&3y&=&4\\5x&+&4y&=&17\end{matrix}$$
Notes
There are 3 row operations we can perform on a matrix without changing the solution of the system:
Interchange rows
Replace a row by a non-zero multiple of itself
Replace a row by adding a multiple of the row to a multiple of another row.
We are aiming to get to something of the form$$\left(\begin{array}{cc|c}1&0&a\\0&1&b\end{array}\right)$$for there to be a single unique solution. This boils down to \(1x=a\) and \(1y=b\).
If you get something of the form $$\left(\begin{array}{cc|c}1&0&a\\0&0&b\end{array}\right)$$where \(b\ne 0\) then there are no solutions. This is because the second row says \(0x+0y=b\) which is impossible if \(b\ne 0\).
If you get something of the form $$\left(\begin{array}{cc|c}1&b&a\\0&0&0\end{array}\right)$$ then there are infinitely many solutions. The first row gives \(x+by=a\), but there is no second equation now to find a single solution.
The same method applies to a system of three linear simultaneous equations in three unknowns.
Examples and Your Turns
Example
Solve this pair of simultaneous equations using matrix row reduction $$\begin{matrix}2x&+&4y&+&z&=&5\\3x&-&5y&-&z&=&4\\x&+&y&-&z&=&6\end{matrix}$$
Example
Solve this pair of simultaneous equations using elimination$$\begin{matrix}2x&+&4y&+&z&=&5\\3x&-&5y&-&z&=&4\\x&+&y&-&z&=&6\end{matrix}$$
Example
Solve this pair of simultaneous equations using substitution$$\begin{matrix}2x&+&4y&+&z&=&5\\3x&-&5y&-&z&=&4\\x&+&y&-&z&=&6\end{matrix}$$
Your Turn
Solve this pair of simultaneous equations using a method of your choice$$\begin{matrix}x&+&3y&-&2z&=&3\\2x&-&4y&+&3z&=&5\\4x&+&y&-&z&=&6\end{matrix}$$
Your Turn
Solve this pair of simultaneous equations using a method of your choice$$\begin{matrix}x&+&2y&+&z&=&3\\2x&-&y&+&z&=&8\\3x&-&4y&+&z&=&18\end{matrix}$$
Your Turn
Solve this pair of simultaneous equations using a method of your choice$$\begin{matrix}2x&-&y&+&z&=&5\\x&+&y&-&z&=&2\\3x&-&3y&+&3z&=&8\end{matrix}$$
Notes
As was seen in the last two Your Turns, there are sets of three simultaneous linear equations that have no solutions and others that have infinitely many solutions.
I recommend the Matrix Row Reduction method to solving these systems of equations, as it is much easier to determine which situation we are in through clear rules, whereas both elimination and substitution leave it open to a correct interpretation at the end.
For matrix row reduction, if you end up with something of the form$$\left(\begin{array}{ccc|c}1&0&0&a\\0&1&0&b\\0&0&0&c\end{array}\right)$$where \(c\ne 0\) then there are no solutions. This is because the third row says \(0x+0y+0z=c\) which is impossible if \(c\ne 0\).
If you end up with something of the form$$\left(\begin{array}{ccc|c}1&0&0&a\\0&1&0&b\\0&0&0&0\end{array}\right)$$ then there are infinitely many solutions. This is because the third row says \(0x+0y+0z=0\) which is true for infinitely many different values of \(x,y,z\).
Key Facts
Use this applet to generate a prompt for a Key Fact that you need to know for the course. The idea is that you should KNOW these key facts in order to be able to solve problems.