1. Name: Ma. Angeli Oguan BSHM1H Subject: MathOne1
Solving simultaneous equations
The terms simultaneous equations and systems of equations refer to conditions
where two or more unknownvariables are related to each other through an equal
number of equations. Consider the following example:
For this set of equations, there is but a single combination of values for x and y that will
satisfy both. Eitherequation, considered separately, has an infinitude of
valid (x,y) solutions, but together there is only one. Plotted on a graph, this condition
becomes obvious:
Each line is actually a continuum of points representing possible x and y solution pairs
for each equation. Each equation, separately, has an infinite number of ordered pair
(x,y) solutions. There is only one point where the two linear functions x + y = 24 and 2x -
y = -6 intersect (where one of their many independent solutions happen to work for both
equations), and that is where x is equal to a value of 6 and y is equal to a value of 18.
Usually, though, graphing is not a very efficient way to determine
the simultaneous solution set for two or more equations. It is especially impractical for

2. Name: Ma. Angeli Oguan BSHM1H Subject: MathOne1
systems of three or more variables. In a three-variable system, for example, the solution
would be found by the point intersection of three planes in a three-dimensional
coordinate space -- not an easy scenario to visualize.
Substitution method
Several algebraic techniques exist to solve simultaneous equations. Perhaps the
easiest to comprehend is the substitution method. Take, for instance, our two-variable
example problem:
In the substitution method, we manipulate one of the equations such that one variable is
defined in terms of the other:
Then, we take this new definition of one variable and substitute it for the same
variable in the otherequation. In this case, we take the definition of y, which is 24 - x and
substitute this for the y term found inthe other equation:
Now that we have an equation with just a single variable (x), we can solve it using
"normal" algebraic techniques:

3. Name: Ma. Angeli Oguan BSHM1H Subject: MathOne1
Now that x is known, we can plug this value into any of the original equations and obtain
a value for y. Or, to save us some work, we can plug this value (6) into the equation we
just generated to define y in terms ofx, being that it is already in a form to solve for y:
Applying the substitution method to systems of three or more variables involves a
similar pattern, only with more work involved. This is generally true for any method of
solution: the number of steps required for obtaining solutions increases rapidly with
each additional variable in the system.

4. Name: Ma. Angeli Oguan BSHM1H Subject: MathOne1
To solve for three unknown variables, we need at least three equations. Consider this
example:
Being that the first equation has the simplest coefficients (1, -1, and 1, for x, y, and z,
respectively), it seems logical to use it to develop a definition of one variable in terms of
the other two. In this example, I'll solve for x in terms of y and z:
Now, we can substitute this definition of x where x appears in the other two equations:
Reducing these two equations to their simplest forms:

5. Name: Ma. Angeli Oguan BSHM1H Subject: MathOne1
So far, our efforts have reduced the system from three variables in three equations
to two variables in twoequations. Now, we can apply the substitution technique again to
the two equations 4y - z = 4 and -3y + 4z = 36 to solve for either y or z. First, I'll
manipulate the first equation to define z in terms of y:
Next, we'll substitute this definition of z in terms of y where we see z in the
other equation:

6. Name: Ma. Angeli Oguan BSHM1H Subject: MathOne1
Now that y is a known value, we can plug it into the equation defining z in terms of y and
obtain a figure forz:
Now, with values for y and z known, we can plug these into the equation where we
defined x in terms of yand z, to obtain a value for x:

7. Name: Ma. Angeli Oguan BSHM1H Subject: MathOne1
SIMULTANEOUS EQUATION BY ELIMINATION
Theory:
In the ‘elimination’ method for solving simultaneous equations, two
equations are simplified by adding them or subtracting them. This
eliminates one of the variables so that the other variable can be
found.
To add two equations, add the left hand expressions and right hand
expressions separately. Similarly, to subtract two equations, subtract
the left hand expressions from each other, and subtract the right
hand expressions from each other. The following examples will make
this clear.
Example 1: Consider these equations:
2x 5y = 1
3x + 5y = 14
The first equation contains a ‘ 5y’ term, while the second equation
contains a ‘+5y’ term. These two terms will cancel if added together,
so we will add the equations to eliminate ‘y’.
To add the equations, add the left side expressions and the right side
expressions separately.

8. Name: Ma. Angeli Oguan BSHM1H Subject: MathOne1
2x 5y = 1
+ 3x + 5y = +14
(2x 5y) + (3x + 5y) = 1 + 14
Simplifying, 5y and +5y cancel out, so we have:
5x = 15
Therefore ‘x’ is 3.
By substituting 3 for ‘x’ into either of the two original equations we
can find ‘y’.
Example 2:
The elimination method will only work if you can eliminate one of
the variables by adding or subtracting the equations as in example
1 above. But for many simultaneous equations, this is not the case.
For example, consider these equations:
2x + 3y = 4
x 2y = 5
Adding or subtracting these equations will not cancel out the ‘x’ or
‘y’ terms.
Before using the elimination method you may have to multiply every
term of one or both of the equations by some number so that equal
terms can be eliminated.
We could eliminate ‘x’ for this example if the second equation had a
‘2x’ term instead of an ‘x’ term. By multiplying every term in the
second equation by 2, the ‘x’ term will become ‘2x’, like this:

9. Name: Ma. Angeli Oguan BSHM1H Subject: MathOne1
x 2 2y 2 = 5 2
giving:
2x 4y = 10
Now the ‘x’ term in each equation is the same, and the equations
can be subtracted to eliminate ‘x’:
2x + 3y = 4
2x 4y = 10
(2x + 3y) (2x 4y) = 4 10
Removing the brackets and simplifying, the ‘2x’ terms cancel out, so
we have:
7y = 14
So
y=2
The other variable, ‘x’, can now be found by substituting 2 for ‘y’ into
either of the original equations.
Sometimes both equations must be modified in order to cancel a variable. For
example, to cancel the ‘y’ terms for this example, we could multiply the first
equation by 4, and the second equation by 3. Then there would be a ‘12y’ term
in the first equation and a ‘-12y’ term in the second equation. Adding the
equations would then eliminate ‘y’.