# Solving simultaneous equations graphically

**Simultaneous equations**

There are circumstances where equations in two unknowns have an infinite number of solution pairs. For example,

*x* + *y* = 3 therefore;

when *x* = 1, *y* = 2

when *x* = 3, *y* = 0

when *x* = -2, *y* = 5

This can be graphically presented as follows:

The equation *y* – *x* = 1 can also be added to the graph and will also have an infinite number of solutions:

For the equation* x* + *y* = 3 and *y* – *x* = 1 there is one pair of values which will solve them both. This is the point where the two lines intersect (1, 2).

Therefore at this point *x* = 1 and *y* = 2.

*x* + *y* = 3 and *y* – *x* = 1 are called **simultaneous equations**.

**Simultaneous equations with no solutions**

Two equations which product graphs which are parallel can also be simultaneous equations. However, the lines never meet so there is no point of intersection, this means that there are no solutions.

One of the ways to know if two lines are parallel is if they have the same gradient. This can be worked out by rewriting the linear equation in the following form:

*y* = *mx* + *c*

For example:

*y* – 2*x* = 3

2*y* = 4*x* + 1

Rearranging in the form *y* = *mx* + *c *becomes:

*y = *2*x + 3*

*y* = *2**x* + 0.5

The gradient for both equation is 2, (*m* = 2) therefore there are no solutions.

**Simultaneous equations with infinite solutions**

There also maybe circumstances where a pair of simultaneous equations could be represented by identical graphs. In this case, they both have an infinite number of solutions.

They only way to check this is to re-arrange both formulas in the *y* = *mx* + *c *form.