Review question

# Can we solve these simultaneous equations of degree 1 and 2? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R6797

## Solution

Solve the simultaneous equations $\frac{x}{3} - \frac{y}{10} = \frac{5}{6}, \quad x(y-2) = 2y + 3.$

Let’s start by multiplying the first equation by $30$, which is the lowest common multiple of the denominators; in this way, we will have no more fractions. The equation thus becomes $10x-3y=25.$

We could rearrange this to find $x = \dfrac{3y+25}{10}$ and substitute this into the second equation, but this reintroduces the fractions and so seems somewhat ugly.

Alternatively, we could rearrange the second equation to get $x=\dfrac{2y+3}{y-2}$ and substitute this into the first equation, which we can then rearrange and solve to find $y$. This, though, also has fractions in it, and this time they are algebraic.

A somewhat nicer approach is to multiply the first equation by $y-2$, to get $10x(y-2)-3y(y-2)=25(y-2).$ Since we know $x(y-2)=2y+3$, this now becomes $10(2y+3)-3y(y-2)=25(y-2)$, which we can rearrange to get $3y^2-y-80=0.$

This equation factorises into $(y+5)(3y-16)=0$, thus $y = -5$ or $y=\dfrac{16}{3}$. (Alternatively, we could use the quadratic formula to obtain these solutions.)

The corresponding values for $x$ (using either of the original equations or their rearranged variants) are $x=1$ and $x=\dfrac{41}{10}$. We check that our answers are correct by substituting these $(x,y)$ pairs into the other equation.

Therefore there are two solutions: $(x,y) = (1, -5)$ or $(x,y)= \left(\dfrac{41}{10}, \dfrac{16}{3}\right)$.

Looking at this geometrically, $10x-3y=25$ is a straight line which intersects the axes at $(\frac52,0)$ and $(0,-\frac{25}{3})$, while we can rearrange $x(y-2) = 2y + 3$ to get $y=\frac{2x+3}{x-2}=\frac{2(x-2)+7}{x-2}=2+\frac{7}{x-2}$ so this is a stretched and translated $y=\frac{1}{x}$ graph (which is a rectangular hyperbola).

The straight line and other graph intersect twice, as we have discovered: