Post a New Question | Current Questions | Live Experts
Homework Help: Mathematics: Algebra: Simultaneous Equations
by Rik Hewis
What is a simultaneous equation?
An algebraic equation is one which contains an unknown term. This is rpresented by a letter, often, but not always the letter x. This letter can stand for any value and it’s usually a simple process to work out what it represents. For instance this can be done as follows:
5x + 3x + 4 = 4x + 3
8x + 4 = 4x + 3
8x - 4x = -1
4x = -1
x = -0.25
Here, we have simplified the expression by gathing all the number terms and the terms in x, and simplifying them. Then it is a simple matter to simplify this to the point where we can give a solution for x.
However, what if out equation is not in just x? For instance, we may have an equation in terms of x and y, such as:
The solution could be x=3 and y=1. But then again, it could be any solution, and for a given x, a y can be found. However, we can fix the values of x and y, by adding a second equation:
3x + 4y = 13
Now we can see that if x=3 and y=1, then both equations are satisfied, but if we try a different value that satisfies one equation (eg x=4, y =0.25), then we can see that this does NOT satisfy the second equation.
So, how do I solve a set of simultaneous equations?
Take the set of simultaneous equations:
2x + 4y = 28
3x - 6y = 18
This can be solved in a number of ways.
Firstly we consider substitution.
Substitution
We have two equations with two unknowns, x and y. But we know how to solve equations with one unknown, so if we could get rid of one unkown, then we could easily solve this.
Take one equation, say the first one. Now attempt to solve this for y (or x) in the same way as we did for a single unkown before:
2x + 4y = 28
4y = 28 - 2x
y = (28 - 2x) / 4
We said before that this isn’t much use, as it won’t give us a value for just y, ie, our value for y has x in it, which is also an unknown. But remember now we have a second equation, containing x and y. Also, we know what y is, in terms of x. If we put our values of x into the other equation, then we find that suddenly we’ve got one equation, which, while it might seem a little messy, is fairly straight forward to rearrange.
3x - 6y = 18
y = (28 - 2x) / 4
3x - 6((28 - 2x) / 4) = 18
3x - 42 + 3x = 18
6x = 60
x = 10
Now we only have one unknown, y. If we put our value of x into the other equation, then we can easiy calculate y, like we did before for single-unknown equations.
2x + 4y = 28
x = 10
20 + 4y = 28
4y = 28 - 20
y = 8 / 4 = 2
Elimination
Consider the statement:
(3x + 4y) - (x + 4y)
We can simplify this by actually doing the sum (This is actually identical to what we did before when we grouped the x terms and the y terms togther). We get:
(3x - x) + (4y - 4y)
2x - 0y
But hang on! If we have 0y, then we have 0 (as zero multiplied by any number is zero), so we can disregard it. The y term has gone (we say it has been eliminated).
How does this help us in solving our simultaneous equation? The x and y terms might not neatly subtract (or add in the case of negative numbers).
Think back, though, to when we are rearranging equations to solve for a single equation. If we have , then we can rearrange this to solve it. One way of visualizing the simplification is to divide both sides of this equation by the same number. This can be any number as long as both sides are divided by the same number.
We can do exactly the same thing with a simultaneous equation. Take the set of equations:
4x + 7y = 6
3x - 2y = 19
We can then say we want to eliminate y (for instance). However we can clearly see that the terms in y are not equal, so subtracting one from the other will not help. However, if we multiply the first one by 2 and the second one by 7, (note: Observant readers will have noticed that these numbers are the co-efficients of y from the other equation)
4x + 7y = 6 Multiply both sides by 2 to get:
8x + 14y = 12
3x - 2y = 19 Multiply both sides by 7 to get:
21x - 14y = 133
When we write one above the other we can see clearly that we can add the equations and the y terms will vanish.
8x + 14y = 12
21x - 14y = 133 Add like terms to get:
29x + 0y = 145
It's now a simple process to solve for x, and then use this value to solve for y.
Homework Help: Mathematics: Algebra
For Further Reading
|
