AS91587

Simultaneous Equations

• In mathematics, a system of linear equations (or linear system) is a collection of linear equations involving the same set of variables.

• A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and (the first power of) a single variable.

This is a system of linear equations

This is not a system of linear equations

• is a linear system of three equations in the three variables x, y, z.

Solving equations

• There is a unique solution as -7 is the only value of x that makes the LHS = RHS

We don’t always get a solution though

• If we try to solve the following:

• We find that this is not true and hence there are no solutions.

Solving 2D systems

• Understanding the equations:

• Example:

• This equations has 2 variables and there are an infinite number of solutions i.e.

Every point on this line satisfies the equation so there are an infinite number of solutions.

e.g. (0, 3), (1,5), (0.2,3.4) are some of these solutions

We can only get a solution for ‘y’ if we know the particular value of ‘x’ i.e. ‘x’ is no longer a variable.

2 equations with 2 variables each

• Solve

Solving using substitution

• Solve

Solving using elimination

There is only one point that lies on both lines and so this point (1, 5) is a unique solution

Example

• Solve:

Example

There is a unique solution (1, 3)

If the lines have different gradients, they must intersect and give us a unique solution.

Example

• Solve:

Example

• This is not possible

The lines will never intersect and hence there is no solution

Notice that the LHS is the same but the RHS is different

• This means the lines have the same gradient but are separated.

But if the LHS is the same as the RHS, then every point matches and hence there are an infinite

number of solutions

• This means the lines have the same gradient but are separated.

It could look like this

• The second line is a multiple of the first line

Summary

• Left hand sides have different gradients so we expect a unique solution

The systems of equations are consistent with a unique solution

Summary

• Left hand sides have the same gradients and the right hand sides are different so we expect no solution

No solutions. The system of equations is inconsistent.

Summary

• Left hand sides have the same gradients and the right hand sides are in proportion so we expect infinite solutions

One line matches the other line exactly and so they have infinite solutions.

The system of equations is consistent with infinite solutions