Simultaneous Equations

Have you heard simultaneous equation? How can we solve if the equation has two unknown value?
The simultaneous equation is an equation with two or more unknown value which the value of the unknown is same to the other equation. There is a three method to find the value of unknown which are the substitution, elimination, and by graphing.

Using the given 2x = y + 7 and x + y = 2, let’s solve for the unknown using substitution, elimination, and by graphing. First, the name the following equation as [1] and [2]. It will be:
[1] 2x = y + 7
[2] x + y = 2

Substitution Method
Substitution method with the given two equation we will choose one equation to form “x=” or “y=” then use the revised equation to solve for the missing value.

Steps to Solve by Substitution Method
First, from the equation, you choose. We only need to have one variable to the left side of the equal sign to form “x=” or y=”. In our example above, you can choose to use any of the two equations. For this, we will use the second equation [2] x + y = 2.

Let’s have “x =” and we will move y to the other side.
[2] x + y – y = 2 – y subtract y to each side then the equation will be x = 2 – y.

Second step is to substitute x = 2 – y to the first equation which is [1] 2x = y + 7 it will means that the value of x is 2 – y. It will be:
2( 2 – y) = y + 7 Using Distributive property it will be
2(2) – 2(y) = y + 7
4 – 2y = y + 7 Combined similar term
-2y – y = – 4 + 7
-3y = 3 divide both side by -3
y = -1 this is the value of y and to get value of x substitute the value of y to any of the two equations.

The third step, solve for the other variable. Here, we will be using the first equation.
2x = y + 7 Substitute -1 to y
2x = -1 + 7
2x = 6 divide both sides by 2
x = 3

Note: To check your answer, choose any equation and substitute both values.

Elimination Method
In elimination method, using the two formula to remove one variable. You can choose any variable then proceed to substitution. Using the given [1] 2x = y + 7 and [2] x + y = 2. Arrange the first equation, to be similar to the second equation 2x – y = 7 then proceed to the method of elimination.

We can easily compute if we align the two equation. Here we have
2x – y = 7
x + y = 2 add the two equation
3x + 0 = 9 Divide both sides by 3 to get the value of x
x = 3

We can get the value of y by substituting the value x. Choose any of the two equation then substitute for the value of x.
[2] x + y = 2
3 + y = 2 combined similar term subtract both sides by -3. It will be -3 + 3 + y = 2 – 3
y = 2 – 3
y = -1

Note: In elimination, arrange equation to the standard form of linear equations in two variable or ax + by = c.

Graphing method
The graphing method is a plot line or lines that represents a linear equation. In solving using graph usually, we extend the line to know where the one point where the two lines, two equations, intersect.

To graph an equations, simply substitute if the x = 0 or y = 0. Let’s have the
[1] 2x = y + 7
[2] x + y = 2
Note: We only used a low number such as 0, 1, 2 3 or any number for x or y to easily compute the corresponding value.

[1] 2x = y + 7
Let’s solve using the first equation [1] 2x = y + 7 or the standard form which is 2x – y = 7. Using 2x – y = 7 let x = 0. Substitute 0 to x.
2(0) – y = 7
-y = 7 divide both sides by -1
y = -7
If x = 0 then (0 , -7)

Using the same equation let y = 0
2x – 0 = 7
2x = 7 divide both sides by 2
x = 7/2 or 3 1/2
If y = 0 then (3 1/2 , 0)

Therefore, in the first equation, 2x – y = 7, the set of coordinates are (0 , -7) and (3 1/2, 0)

[2] x + y = 2
Do the same for the second equation.
x = 0
0 + y = 2
y = 2
The set of points is (0 , 2)

y = 0
x + 0 = 2
x = 2
(2 , 0)

For the second equation, x + y = 2, the set or coordinates are (0 , 2) and (2 , 0)

Plot the points of the two equations on the same graph. In the given graph the two lines intersect at (3 , -1).