Simultaneous Equations

What are simultaneous equations?


Please make sure that you know how to rearrange equations, solve for x in linear equations and can substitute. Simultaneous equations are not hard to do if you know how to do these things, it is just a matter of understanding why we solve simultaneous equations.
Suppose we have two linear equations with 2 unknowns which are the same in both equations. We need to find the value of those 2 unknowns which satisfy both equations simultaneously (at the same time).
Question 1:
Find the values of x and y which satisfy both linear equations simultaneously, use the elimination method and substitution method.
  1. x + y = 5
  2. 2x + y = 12

Above are two linear equations, we need to find the values of x and y which satisfy both equations. What do we mean by satisfy both equations? It means that when we find the values for x and y and substitute those values into both linear equations, both sides of each linear equation would be equal.
If x = 2 and y = 3
The first linear equation would be satisfied as
  1. 2 + 3 = 5

If we substitute x = 2 and y = 3 into the second linear equation
  1. 2(2) + 3 = 7 ≠ 12 (WE NEED IT TO EQUAL 12 NOT 7)

As a result we know that the values of x and y cannot be x = 2 and y = 3, as the SECOND EQUATION IS NOT SATISFIED, WE NEED TO FIND THE VALUES OF X AND Y WHICH SATISFY BOTH EQUATIONS.
How do we do this, we can use two METHODS, one is called the Substitution method and the other is called the Elimination method.

Substitution Method


The substitution method chooses one of the linear equations, then makes one of the unknown numbers represented by letters the subject. The expression which the subject equals is substituted into the second linear equation. YOU MAY NOT UNDERSTAND THIS AT FIRST BUT LOOK BELOW AND SEE HOW WE DO THIS, REMEMBER YOU MUST KNOW HOW TO REARRANGE EQUATIONS AND SUBSTITUTE.
Lets choose linear equation 1 which is x + y = 5We need to make either x or y the subject, lets CHOOSE and make x the subject.
x + y = 5
Subtract y from both sides
x + y - y = 5 - y
x = 5 - y
Thus, we know what x equals. We can now substitute x with (5 - y) into the second equation, as this is what equals x.
Remember the second equation is
2. 2x + y = 12
We know what x equals, as x = 5 - y, so we can substitute for x in the second linear equation with (5 - y), as shown below.
2(5 - y) + y = 12 (LETS EXPAND AND SIMPLIFY)
10 - 2y + y = 1210 - y = 12 (We have collected like terms)
So now we have the linear equation 10 - y = 12.
Remember all we have done is find the equivalent of x using the first linear equation. As a result of finding the equivalent, we know what x equals, and thus this can substituted into the second equation to get 10 - y = 12. What is the value of y? Well, we can solve for y, as shown below.
10 - y = 12
Add y from both sides
10 - y + y = 12 + y
10 = 12 + y
Subtract 12 from both sides
10 - 12 = y
-2 = y (WE HAVE OUR VALUE FOR y)
So now we know our value for y which is -2. Given we know the value for y, we can substitute the value of y into ANY linear equation, either 1 or 2 and find what x equals, lets CHOOSE the first equation for the substitution.
The first linear equation is x + y = 5, if y = -2, then solve to find x
x + -2 = 5x - 2 = 5 Add 2 to both sides
x - 2 + 2 = 5 + 2
x = 7 (WE HAVE OUR VALUE FOR x)
Given x = 7 and y = -2, lets check whether both linear equations are satisfied when we substitute the values for x and y IN BOTH LINEAR EQUATIONS.
  1. x + y = 5
  2. 2x + y = 12

Lets substitute the values
  1. 7 + -2 = 7 -2 = 5 YES THIS EQUATION IS SATISFIED
  2. 2(7) + -2 = 14 - 2 = 12 YES THIS EQUATION IS SATISFIED

So our solution is x = 7 and y = -2.
Another method we can use to find the values of x and y is by using the elimination method.

Elimination Method


Lets now solve the same question, however this time using the elimination method.
Question:
Find the values of x and y which satisfy both linear equations simultaneously.
  1. x + y = 5
  2. 2x + y = 12

The idea of the elimination method is to subtract or add one linear equation from the other to eliminate either one of the unknown numbers represented by letters. This would leave us with a linear equation which we need to solve.
The idea is to have the same number of x's or y's in both equations, to eliminate either x or y.
REMEMBER THAT WHATEVER WE DO TO ONE SIDE OF THE EQUATION, WE HAVE TO DO TO THE OTHER.
On the left hand side of the first linear equation there is a 1x term and on the left hand side of the second linear equation there is a 2x term, thus the number of x's are not equivalent. However, in our two linear equations we have the same number of y terms on the left hand side of the equations, so we can subtract one equation from the other.
Lets subtract equation 1 from equation 2. BEAR IN MIND WE ARE STILL TREATING EACH SIDE EQUALLY, AS SHOWN BELOW
2x + y = 12
Subtract the first equation (x + y = 5) from both sides
2x + y - (x + y) = 12 - (x + y)
2x + y - x - y = 12 - (5)
Remember x + y is equal to 5, so on the right hand side of the equation we can write either (x + y) or 5 as they are both equal, we need to write (x + y) on the left hand side, otherwise we would not be able to get rid of the y term. If we did not write 5 and instead wrote x + y, even though they are the same, we would not be getting rid of the y term, as it would end up on the right hand side. Thus our aim is to ELIMINATE one term to get a linear equation where we solve for one term.
2x - x = 12 - 5x = 7 (We have found our value for x)
Now, as normal, given we have found our value for x, we can substitute the value for x in either linear equation and then solve to attain y.
Lets solve to find y now by substituting x with 7 in the first linear equation.
7 + y = 5
Subtract 7 from both sides
7 + y - 7 = 5 - 7
y = -2 (We have found our value for y)
So either the elimination or substitution method can be used, they would give the same answers.

Video 1 Explanation (Elimination method)


Question 1: Solve the simultaneous equations below, ONLY by using the elimination method.
  1. 2x + 4x + 2y = 45
  2. -3x + 4y = 34

We can clearly see that equation 1 can be simplified by collecting like terms, remember, 2x + 4x + 2y = 6x + 2y, so we can write equation 1 and equation 2 again with the simplified equation 1
  1. 6x + 2y = 45
  2. -3x + 4y = 34

We need to eliminate either the x or y term. Lets choose to eliminate the x term; we can do this when the amount of x terms in the left hand sides of both equations are the same or if there ax terms in one equation there can be -ax terms in the second equation, where a is number. At the moment, the left hand side of equation 1 has 6x and the second equation has -3x. We need to choose what amount of x's should each equation have, so we can add or subtract them.
Lets say we want both equations to have 6x, already equation 1 has this, so we do nothing to equation 1.Equation 2 on the other hand needs to be manipulated to get 6x on the left hand side, so we have to multiply BOTH SIDES OF EQUATION 2 BY -2. Remember whatever we do to one side, we have to do to the other, so that both sides remain equal.
-3x + 4y = 34
Multiply both sides by -2
-2(-3x + 4y) = -2(34)
6x -8y = -68
Given we have this equation now we can subtract this equation from this equation 1, as shown below. 6x - 8y = -64 is just both sides of equation 2 multiplied by -2, the equality has remained.
  1. 6x + 2y = 45

Subtract 6x - 8y from both sides
6x + 2y - (6x - 8y) = 45 - (6x - 8y)
6x + 2y - 6x + 8y = 45 - (-68)
(REMEMBER THAT 6x - 8y IS EQUAL TO -68, so we can replace (6x -8y) on the right hand side with -68 as we are trying to get rid of the x terms)
10y = 45 + 68
10y = 113
Divide both sides by 10
10y/10 = 113/10
y = 11.3
Given we now know what y equals we can substitute the value of y into any equation we had before, lets choose equation 2.
  1. -3x + 4y = 34

Lets substitute y for 11.3 to find x.
-3x + 4(11.3) = 34
-3x + 45.2 = 34
Subtract 45.2 from both side
-3x + 45.2 - 45.2 = 34 - 45.2
-3x = -11.2
Divide both sides by -3
-3x/-3 = -11.2/-3
x = 11.2/3 (Answer is in fraction form)
ONE IMPORTANT THING TO MENTION IS that we chose to get 6x on both sides by multiplying only the second equation by -2, due to this we can subtract one equation from another and eliminate the x term. What if we multiplied both sides of equation two by 2, we would have got
2. -3x + 4y = 34
Multiply both sides by 2
2(-3x + 4y) = 2(34)
-6x + 8y = 68
So this equation would have to be ADDED to equation 1. Why add? Remember we need to eliminate the x term.
So
equation 1. 6x + 2y = 45
add (-6x + 8y) to both sides
6x + 2y + (-6x + 8y) = 45 + (-6x + 8y)
6x + 2y - 6x + 8y = 45 + (68)
10y = 45 + 68
10y = 113
Divide both sides by 10
10y/10 = 113/10
y = 11.3 (We get the same answer, go check)
Lets now substitute the value for y into any equation, lets choose equation 1.
  1. 6x + 2y = 45

6x + 2(11.3) = 45
6x + 22.6 = 45
Subtract 21.8 from both sides
6x = 45 - 21.8
6x = 22.4
Divide by 6
6x/6 = 22.4/6
x = 22.4/6 = 11.2/3 (We get the same answer, go check)