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1234Linear relations

## Theory

Take a look at the applet: Linear Relations

An equation of the form $ax+by=c$ describes a linear relationship with two variables. The corresponding graph is a straight line. You can find this graph by determining points along this line. This is often done by choosing $x=0$ and calculating the corresponding value for $y$, and by then choosing $y=0$ and calculating the corresponding value for $x$.

Linear equations with two variables such as $ax+by=c$ can be rewritten as a linear function as long as $b\ne 0$.
For example the equation $2x+3y=6$ can be reorganised to give $y=-\frac{2}{3}x+2$.
This is then a linear function with y -intercept $2$ and gradient $-\frac{2}{3}$.

Special cases:

• $a=0$: the equation then takes on the form $by=c$ and can be rewritten as $y=\frac{c}{b}$. This is a linear function with gradient $0$.

• $b=0$: the equation takes on the form $ax=c$ and can be rewritten as $x=\frac{c}{a}$. This is not a linear function as there is no gradient. The graph is a line parallel to the $y$ -axis.