What Simultaneous Equations Mean

Simultaneous equations are two equations that are both true at the same time. The solution is the pair of values \( (x, y) \) that satisfies both equations.

 

When you solve them by graphical methods, you draw both straight lines on the same axes then find the point where they intersect.

 

 

Forming the Two Lines

Each equation should be in the form \( y = mx + c \) so it is easy to plot.

 

Example equations

$$
2x + y = 8
$$

$$
x - y = 1
$$

 

Rearrange each one to make \( y \) the subject.

 

First equation:

$$
2x + y = 8
$$

$$
y = 8 - 2x
$$

 

Second equation:

$$
x - y = 1
$$

$$
-y = 1 - x
$$

$$
y = x - 1
$$

 

 

Plotting Each Line

To draw a straight line, you need at least two points. A simple method is to choose whole number \( x \) values and calculate \( y \).

 

For \( y = 8 - 2x \), choose \( x = 0 \) and \( x = 4 \).

 

$$
x = 0
$$

$$
y = 8
$$

 

$$
x = 4
$$

$$
y = 0
$$

 

So two points are \( (0, 8) \) and \( (4, 0) \).

 

For \( y = x - 1 \), choose \( x = 0 \) and \( x = 2 \).

 

$$
x = 0
$$

$$
y = -1
$$

 

$$
x = 2
$$

$$
y = 1
$$

 

So two points are \( (0, -1) \) and \( (2, 1) \).

 

Draw both lines on the same axes using a ruler.

 

 

Finding the Solution from the Intersection

The solution is the coordinates of the point where the two lines cross.

 

From the graph, read the intersection point.

 

In this example, the lines intersect at:

$$
(3, 2)
$$

 

So the solution is:

$$
x = 3
$$

$$
y = 2
$$

 

 

Checking the Solution

It is good practice to check by substitution.

 

Check in \( 2x + y = 8 \):

$$
2(3) + 2 = 8
$$

$$
8 = 8
$$

 

Check in \( x - y = 1 \):

$$
3 - 2 = 1
$$

$$
1 = 1
$$

 

Both are true so the solution is correct.

 

 

Key Points to Remember

Rearrange both equations into \( y = mx + c \) before plotting.
Plot at least two points for each line then join them with a ruler.
The solution is the intersection point \( (x, y) \) read from the graph.
Graphical solutions can be approximate so read carefully and use a sensible scale.
Always substitute the intersection values back into both equations to check.

 

Graphical methods solve simultaneous linear equations by turning each equation into a line and using their intersection to find the common solution.