Simultaneous Equations (w.Graphs)In maths, a simultaneous equation simply means 2 equations where the values for x and y are the same.

A really simple example could be:

y=x+2and y=x2 +2

If y = 3 and x = 1, then both equations are true:

3 = 1 + 2 and 3=12 + 2 - so y=3 and x=1 is a solution to these simultaneous equations!

There are a few ways you can find solutions to simultaneous equations. We are going to look at solving using graphs.

First Example: Linear EquationsThe graphs of linear equations will give straight lines. Solve these simultaneous equations by drawing graphs:

2x + 3y = 6 (plotted in red) 4x - 6y = - 4 (plotted in blue)

Solutions are INTERSECTS

At intersects, the values of x and y are the same.

Second Example: Linear and Quadratic EquationsThe graphs of quadratic equations will give curved lines.Solve the simultaneous equations by drawing graphs.

y - 2x = 1 (plotted in blue) y = x2 - 2 (plotted in red)

Solutions are INTERSECTS

At intersects, the values of x and y are the same.

Yay!

No plotting

Required!!

The quadratics are plotted for

you on the other sheet!

Question PracticeThere are four graphs opposite.

Each is labeled with its equation.

Use the graphs to find the solutions tothese simultaneous equations

Q2 (Medium)Solve the following simultaneous equations:

*You will need to do some rearranging of formulae for d) and f)!!

a) y=x2−2 , y=x+14b) y=x2−8 , y=3x+10c) y=2x2 , y=x+3

d) x+5 y=25 , x2+45x= y

e) y=4−2 x , y=x2+4f) y=3 (x¿¿2+3) ,4 x+ y=1¿

Q3 (Hard)Find the value of x and y for each of the following rectangles by first writing down a pair of simultaneous equations and then solving them:

a)

y=x2−xy=x+2

b)

y=x2−x y=8

c)

y=x2−xy=−2x+4

d)

d)

y=−2x+4y=x+2

e)

y=−2x+4 y=8

f)

y=x+2y=−2x+4

d)

3y + 2x

18

y + 3x

6

12

7

2x – 3y

4y + 5x

13

4x – 6yx + y

2

These look hard, but you

just PLOT, then find the

INTERSECTS.