Solving simultaneous equations using graphs

Solving simultaneous equations using a graph is easier than you might think. First, you need to draw the lines of the equations. The points where the lines cross is the solution.

Linear equations

The graphs of linear equations will give straight lines.

Example

• Solve these simultaneous equations by drawing graphs:
• 2x + 3y = 6
• 4x - 6y = - 4

For example, to draw the line 2x + 3y = 6 pick two easy numbers to plot. One when x = 0 and one where y= 0

• When x = 0 in the equation 2x + 3y = 6
• This means 3y = 6 so y = 2
• So one point on the line is (0, 2)

• When y = 0
• 2x = 6 so x = 3
• So another point on the line is (3 ,0)

In an exam, only use this method if you are prompted to by a question. It is usually quicker to use algebra if you are not asked to use graphs.

Linear and quadratic equation

Example

• Solve the simultaneous equations by drawing graphs.
• y - 2x = 1
• y = x² - 2

How to convert a recurring decimal into a fraction

Example 1

Write 0.77..$\mathrm{0.77..}$ as a fraction in its lowest terms.

Our first step is to form a simple equation where x=0.77..$x=\mathrm{0.77..}$. By multiplying both sides by 10$10$ we can obtain another equation with 10x=7.77..$10x=\mathrm{7.77..}$. Now we eliminate the recurring part of the decimal by subtracting x$x$ from 10x$10x$.

x10x9xx=0.77..=7.77..=7=79$$\begin{array}{rl}x& =\mathrm{0.77..}\\ 10x& =\mathrm{7.77..}\\ 9x& =7\\ x& =\frac{7}{9}\end{array}$$

So we have our answer 0.77...=79$\mathrm{0.77...}=\frac{7}{9}$.
The important part to remember is to get two equations in x$x$ where the recurring part after the decimal point is exactly the same.

Example 2

Write 0.277..$\mathrm{0.277..}$ as a fraction in its lowest terms.

Again our first step is write x=0.277..$x=\mathrm{0.277..}$. Now multiplying by 10$10$ gives us 10x=2.77..$10x=\mathrm{2.77..}$. This time we need another equation to match the recurring part of the equation. So multiplying by 10$10$ again gives 100x=27.77..$100x=\mathrm{27.77..}$. Now we have two equations with the same recurring part we subtract one from the other as before.

x10x100x90xxx=0.277..=2.77..=27.77..=25=2590=518$$\begin{array}{rl}x& =\mathrm{0.277..}\\ 10x& =\mathrm{2.77..}\\ 100x& =\mathrm{27.77..}\\ 90x& =25\\ x& =\frac{25}{90}\\ x& =\frac{5}{18}\end{array}$$

So in its lowest terms 0.277..=518$\mathrm{0.277..}=\frac{5}{18}$.

Example 3

Write 0.0855.. as a fraction in its lowest terms.

As always our first step it to write x=0.0855..$x=\mathrm{0.0855..}$. This time to get two equations with the same recurring part after the decimal point we need to multiply by 100$100$ and then 1000$1000$. This gives us:

100x1000x900xx=8.55..=85.55..=77=77900$$\begin{array}{rl}100x& =\mathrm{8.55..}\\ 1000x& =\mathrm{85.55..}\\ 900x& =77\\ x& =\frac{77}{900}\end{array}$$

This cannot be simplified any further so 0.855..=77900$\mathrm{0.855..}=\frac{77}{900}$.

Example 4

Write 0.0234234.. as a fraction in its lowest terms.

This time the '234$234$' part is the important bit. Now to get two multiples of x with 234 as the recurring part we need to multiply first by 10$10$ and then another 1000$1000$ to move the digits 3 places to the right.

x=0.0234234...10x=0.234234..10000x=234.2349990x=234xx=2349990x=13555$$\begin{array}{r}x=\mathrm{0.0234234...}\\ 10x=\mathrm{0.234234..}\\ 10000x=234.234\\ 9990x=234x\\ x=\frac{234}{9990}\\ x=\frac{13}{555}\end{array}$$

So 0.0234=13555$0.0234=\frac{13}{555}$ as a fraction in its lowest terms.