Download linear equation

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
3x - 2y + z = 4
2x + y - z = 8
x- y+3 z = 2
x + y - z = 10
2x + 3y + z = 12
x - y + 5z = 1
• is a linear system of three equations in the
three variables x, y, z.
Solving equations
2x + 4 = x - 3
x = -7
• 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:
2x - 3 = 2x + 7
0 = 10
• We find that this is not true and hence there
are no solutions.
Solving 2D systems
• Understanding the equations:
• Example:
y = 2x+3
• 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.
y = 2x + 3
e.g. (0, 3), (1,5), (0.2,3.4) are some of these solutions
y = 2x + 3
We can only get a solution for ‘y’ if we know the
particular value of ‘x’ i.e. ‘x’ is no longer a variable.
y = 2x + 3
x=2
y = 2´2+3
=7
2 equations with 2 variables each
• Solve
y = 2x + 3
y = -x + 6
Solving using substitution
• Solve
y = 2x + 3
y = -x + 6
Þ -x + 6 = 2x + 3
1= x
y=5
Solving using elimination
y = 2x + 3 (1)
y = -x + 6 (2)
0 = 3x - 3 (Subtracting (1)-(2))
x =1
y=5
There is only one point that lies on both lines
and so this point (1, 5) is a unique solution
Example
• Solve:
2x + 3y = 11
x+y = 4
Example
2x + 3y = 11 (1)
x + y = 4 (2)´ 2
2x + 2y = 8 (3)
y = 3 (1) - (3)
x =1
There is a unique solution (1, 3)
If the lines have different gradients, they must
intersect and give us a unique solution.
Example
• Solve:
y - 2x = 3
y - 2x = -1
Example
y - 2x = 3
y - 2x = -1
0 = 4 (subtracting)
• 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
y - 2x = 3
y - 2x = -1
• 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
y - 2x = 3
y - 2x = 3
• This means the lines have the same gradient but
are separated.
It could look like this
y - 2x = 3
2y - 4x = 6
• The second line is a multiple of the first line
Summary
• Left hand sides have different gradients so we
expect a unique solution
2x + 3y = 7
x - 5y = -3
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
2x + 3y = 7
4x + 6y = -3
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
2x + 3y = 7
4x + 6y = 14
One line matches the other line exactly and so they
have infinite solutions.
The system of equations
is consistent with infinite solutions