Topic 1 Simultaneous equations
1.1 Linear equations
A linear equation in one variable (or unknown) is an equation of the form \[ax=b,\] where \(a\) and \(b\) are constants and \(a\neq 0\).
Example 1.1 \[9x=12\]
To solve it (meaning to find all numbers which, if substituted for \(x\), yield a true statement), we divide both sides by \(9\): \(x = \frac{12}{9} = \frac{4}{3} = 1.\overline{3} \approx 1.33\) (2 d.p.).
In general, the solution of the equation \(ax=b\) is \(x=\frac{b}{a}\).
A linear equation in two variables \(x\) and \(y\) is an equation of the form \[ax+by=c,\] where \(a,b\) and \(c\) are constants, and both \(a\) and \(b\) are not zero. (Note that if one of them would be zero, then the equation has effectively only one variable.). If \(c=0\), then the equation is called homogeneous.
Example 1.2 \[4x-3y=2\]
We can make \(x\) the subject: \(x=\frac{2+3y}{4}=\frac{1}{2}+\frac{3}{4}y\).
We can make \(y\) the subject: \(y=\frac{2-4x}{-3} = -\frac{2}{3} +\frac{4}{3}x\).
For each value of \(x\) we get one value of \(y\) so that the pair \((x,y)\) satisfies the equation. Also vice versa, for each value of \(y\) we get one value of \(x\).
So the equation \(4x-3y=2\) has infinitely many solutions. Table 1.1 shows some of them.
| x | y |
|---|---|
| -2 | -3.3333333 |
| -1 | -2.0000000 |
| 0 | -0.6666667 |
| 1 | 0.6666667 |
| 2 | 2.0000000 |
Graphing these values (Figure 1.1) we see that they appear to lie on a line.
Figure 1.1: Graphing some solutions of \(4x-3y=2\).
The graph of the set of solutions of a linear equation in two variables is always a straight line.
1.2 Solving a pair of linear equations in two variables
Example 1.3 \[\begin{align} x+2y &= -2 \tag{A} \\ 2x+3y &=1 \tag{B} \end{align}\]
1.2.1 By substitution
| 1 | Express \(x\) from (A): | \(x=-2-2y\) |
| 2 | Substitute into (B): | \(2(-2-2y)+3y=1\) |
| 3 | Multiply out: | \(-4-4y+3y=1\) |
| 4 | Rearrange: | \(y=-5\) |
| 5 | Substitute back into (A): | \(x + 2(-5)=-2\) |
| 6 | Rearrange: | \(x=8\) |
1.2.2 By elimination
| 1 | Multiply (A) by 2: | \(2x+4y=-4\) | Call this (A’) |
| 2 | Subtract (A’) from (B): | \(-y = 5\) | |
| 3 | Rearrange: | \(y=-5\) | |
| 4 | Substitute into (A) (as above) | \(x=8\) |
Extra Note (non-examinable): For “2-by-2” systems like these, the two methods are about the same complexity. However for larger systems of linear equations, the latter leads to a quite effective method called Gaussian elimination, which is also amenable to be implemented as a computer algorithm.
1.2.3 Graphical illustration
The solutions of the equation (A) alone lie on some straight line. Likewise for the set of solutions of the equation (B). The solutions of the system of equations, i.e. both equations simultaneously, are exactly the point(s) where do these two lines intersect. For a sketch for the system (A),(B) above, see Figure 1.2.
Figure 1.2: Solutions of (A) and (B) graphically
Extra Note (non-examinable): This is a general principle. Two lines in a plane can intersect either:
- at a single point, or
- at a line, or
- not at all.
These are exactly the options for the set of solution of a pair of linear equations in two variables. Such a system can have:
- a unique solution, or
- infinitely many solutions, arranged on a line, or
- no solutions.
Going to higher dimensions (more equations and more unknowns), the sets of solutions of linear equations are always “flats” in a higher dimensional space (the dimension corresponds to the number of unknowns).