Tips On Using Substitution

Maths entails the usage of our brain juice in solving problems. It is a good platform for stretching our imagination and creativity by using simple concepts learned to handle seemingly complex maths questions.

Let's look at a "complex" simultaneous equations maths problem, and its way of solving (suggested).

Question:

  ---- (A)

--- (B)

Solve for y and x.

How do you go about it?

Look scary, right?

But like what I said, looks can be deceiving. Use the brain to go around the issue!

Tips: The structure of the simultaneous equations looks similar to the conventional type.
(Conventional type:-
Ax + By = nn
Cx + Dy = kk      )

So what we have to do can be to simply substitute  by m, and  by h (or any variable name).

What we thus convert to is:
  ---- (A)

 --- (B)

Will this simultaneous equations be more comfortable to solve?

Hence, a simple twist to the former mathematical questions can result in a totally familiar situations where we have solve many a times.

Thus, the technique and usefulness of substitution cannot be under-estimated.

It can be powerful at times to reveal a beautiful mathematical expression for user to resolve.

Maths Is Interesting!

Treasure our brain and our thinking. 

:-)

Explanation of the Elimination Method

Solving of Simultaneous equations may require one common technique called "Elimination" method.

From the name, we know that it has to eliminate or remove something from the equations.

The target is one selected variable or unknown in the mathematical equations.

However, when approaching this method, you noticed that it involved the subtraction (or addition) of equations.

The question is "Can equations be subtracted?".
And "What is the real meaning of subtracting equations?"

My answers:-

Yes, equations can of course be subtracted. Equations are like other items, e.g. apples, chairs.

The real meaning of subtracting equations is not that apparent.

The true and desired wish to subtract equations boils down to commonising a certain coefficient of a variable.
With this common coefficient, it will then be able to remove this mathematical unknown.

(It is not really the direct processing of equations, and the magical removal of variable as a result!) 

We commonise the coefficient of the selected variable first before subtracting the equations in order that same items are eliminated.

Hope this clarify some doubts of new learners to simultaneous equations solvers.

Concepts have to be learned upfront without pending questions for complete understanding and smooth follow-up learning in the later stage. Seek to clarify any doubts as far as possible.

It will reduce maths anxiety and allow you to enjoy maths as a result. The reward of clearing any doubts cannot be spelled out in words but through actual working and practice with proper analysis.

I believe you support this notion.

Cheers to maths.

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Simultaneous Equations - Decimal Numbered

In the learning of maths, questions grow challenging as one progress upwards.

One such example is the solving of simultaneous equations.

The easy type:

Solve for x and y using elimination method.
4x + 3y = 10 ---- (1)
3x + 4y = 11 ---- (2)

For the above problem can be solved easily by selecting a coefficient to be commonised.
The post of elimination method is reference here for review.

But moving on (higher) with more challenging maths question ... we may get the below.

0.4x + 0.3y = 1 ---- (A)
3x    +    4y  = 11 ---(B)

What should we do next?

Equation (A) may seems unusual.  It is in decimal form!

But as the blog title claims "Maths Is Interesting!", we should not be worried.

This type of question is actually not new in concept or tricky as it seems.
It is there to test you understanding by being "different".

We have to remove the "catch", which is to change the decimated number to integer.
How we do it here is simply multiplying the coefficients by 10.

This makes equation (A) to be  4x + 3y = 10  (back to the original first set of simultaneous equations at the start of this post.

The above example serves to illustrate the simplicity of changing numbers to suit the condition for easy solving. (Other questions may be multiply by another decimal number, or integer).

Just have a clear mind and a confidence attitude will be enough to allow you to solve most of the maths questions.

Try it and you will believe what I say (or write).

Maths is interesting!

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Simultaneous Equations | Re-write equations

*
Simple simultaneous equation problem comes as 2 straight forward mathematical expressions.

Example 1:

3x + y = 4
x + 2y = 3

But some may come in odd expressions (since life is always the case, which makes learning maths more exciting!)

Example 2:

(12 - x)(1 + y) = 15
(8 - x) (1 + y) = -15

Here you will notice that the unknowns are biased towards one side.

Approach 1:
Multiply the 2 factors to get something like example 1.
Using elimination method, remove one of the unknown.
Solve for the only one unknown left.
Using the result found, compute the other unknown.

Approach 2:  (The focus of this post)
Re-write the expression to make it look simpler.

The example 2 can be re-written into below simpler form:
(12 - x) = 15 / (1 + y)  ===>  (A)
(8 - x) = -15 / (1 + y)   ===>  (B)

Equation (B) can then be seen to be the negative of equation (A).

With the re-writing, we will be visually aided to see another form, a simpler one, of the simultaneous equations.

Moving forward with the solution...

12 - x = -(8 - x) = -8 + x
12 + 8 = 2x
x = 20 / 2 = 10     (ANSWER)
Putting x = 10 back into either equation (A) or (B),

We will get (12 - 10) (1 + y) = 15,  if we select equation (A)
1 + y = 15 / 2 = 7.5
y = 7.5 - 1 = 6.5 (ANSWER)

The solution is not the issue in this post.

The key message here is the technique of "re-writing" the equations to reveal the simplicity of the question.

Maths is not that difficult if you look and think to make it easy.

Cheers!
.

Algebra - Why the need?

In doing math, we are actually trying to use a systematic approach to arrive at an answer. It trains the brain to trouble-shoot problems for a consistent result.

Many a times, problems or questions can be solved through guessing and checking the answers to see if they fit the original questions. It is OK.

But if the answers are guessed wrongly, many iterations of the processes take place. It does sound correct or a better way should be better.

Algebra is therefore developed to handle this issue.

See the below simple question of finding the weight of a cheese.



Here, you can see that by guessing the weight of the cheese and checking it back with the other information, you can finally obtain the correct answer.

Another way is to use algebra. This is a systematic approach that when applied, will give you the correct answer on the first try, as opposed to the guessing method.

In the above example, you can replace the weight of the items by x, y, and z.
By solving the simultaneous equations thus formed, you can easily get the answer to the weight of the cheese.

That is the power of algebra and math. No uneducated guess, and time saving as a merit.

Practice, therefore , with algebra as the need is a necessity in our daily life.

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Simultaneous Equations | Three Unknowns

Solving simultaneous equations can be done with many methods. If you understand the underlying concept of elimination, the process will be simple and fun to use.

In this post, the Elimination method to solve 3 unknowns is presented. In a previous post of this website, the steps to solve simultaneous equations having 2 unknowns using Elimination method can be reviewed by clicking this link.

The concept to solving simultaneous equations with 3 unknowns is the same, except the steps are more, since more equations and unknowns are present. Logical, right?

Elimination means the removal of variables or unknown in the process of solving. Study the approach or concept as it serves a good foundation which can be applied to any quantity of unknowns.

The strategy used in the Elimination method of solving is to choose a random unknown to be removed. This reduces the 3-unknown type of question to a simpler 2-unknown type question that can be easily dealt with. The details are presented below.

Letus begin with an example.

x + 2y + 3z = 10 --- (A)
2x + y + z = 9 --- (B)
3x + y + 2z = 13 --- (C)

Step 1:
Look at the 3 equations and decide which unknown is easy to remove. From equation (B) and (C), we can see that the unknown "y" can be easily eliminated by just subtracting the 2 mentioned equations.

(C) - (B): x + 0 + z = 4 --- (D) ==> unknown "y" disappeared !

Step 2:
Find another pair of equations with the purpose of removing the same unknown in step 1.
In this example, there is no 2 other combinations that allow us to remove the same unknown just by performing adding or subtracting. We need to do something first before we can move on in step 2.

Let's target the use of equation (A) and (B) to remove the "y" unknown. Note this is randomly selected.

We see that by multiplying equation (B) by 2, we get 4x + 2y + 2z = 18 ---(E).
The "2y" is now the same as that in equation (A).

Rewriting for simplification,

x + 2y + 3z = 10 --- (A)
4x + 2y + 2z = 18 ---(E)

(A) - (E): -3x + 0 + z = -8 --- (F)

Step 3:
Now we have obtained 2 new equations with only 2 unknowns.
This is the strategy to reduce the more complex 3-unknown question to the simpler 2-unknown question.

Rewriting the 2 new equations here:
x + 0 + Z = 4 --- (D)
-3x + 0 + z = -8 --- (F)

From here onwards, you can review the steps by clicking the link here. It teaches you how to proceed with solving 2-unknown simultaneous equation using the Elimination method.

Some common mistakes made:

The sign of the variables and numbers in the equation are not done correctly after subtraction of 2 equations. Example of this is can be seen using the above question.

If we subtract equation (F) from (D), we should get 4x + 0 = 12.

The common error is to do the x -3x instead of x -(-3x) = x + 3x. The "-" sign error !

To avoid this type of careless mistake, focus on the sign when doing subtraction of equations.
Just keep your mind clear, and constantly remind yourself of this "slip of the mind" mistake.

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Substitution | Simultaneous Equations

Substitution method is one of the method to solve simultaneous equations.

Elimination method is the other techniquqe.

But, do note, substitution method is not as straight forward as the Elimination method.

Concept of Substitution Method:
We replace or substitute one unknown, converted from another equation, to make only a single unknown appears in the equation. Read further to understand.....

Example of Solution:
X + 2y = 5 ----- (A)
2X + y = 7 ------ (B)

Step 1: Convert equation (A) from x + 2y = 5 to x = 5 - 2y
Step 2: Substitute the x = 5 - 2y into equation (B) ==> 2 (5 - 2y) + y = 7

This gives us a final expression of 10 - 4 y + y = 7 which became -3y = 7 - 10 = -3

This results in y = -3 / -3 = 1 ===> y = 1 (Answer) !

Step 3: Replace y = 1 back into any of the original equations (A) or (B), either one will do.

Let's select equation (A) for computing.

x + 2y = 5 ===> x + 2 (1) = 5 ===> x = 5 - 2 (1) = 5 - 2 = 3 ==> x = 3 (Answer)

We have found the two unknowns x = 3 and y = 1! Eureka!

NOTE:
Substitution method is best used when the set of equations are of different order or when the unknowns are in product (multiplied) form.

Elimination method will not work in this case!

See another example below (different order).

Example:
xy = 4 -------(C)
x + y = 5 ------(D)

Step 1: Convert one of the equation (any one) to have unknown as subject. We select equation (D) and unknown x as subject.

x + y = 5 ===> x = 5 - y -----(E)

Step 2: Substitute the new X of equation (E) into equation (C)
xy = 4 ===> (5 - y) y = 4 ===> y^2 - 5y + 4 = 0 (a quadratic equation)

Step 3: Solving the quadratic equation , in this instance, using factorisation method,
yields (y - 4) (y - 1) = 0.

Therefore, from above (y - 4) = 0 or (y - 1) = 0.
Answer: y = 4 or y = 1

Step 4: Replace answer for y into any of original equations

Select equation (D):
When y = 4, ==> x + y = 5 ==> x + 4 = 5 ===> x = 1 (Answer)
When y = 1, ==> x + y = 5 ==> x + 1 = 5 ===> x = 4 (Answer)

From the example shown, it needs more indirect steps compared to the simpler direct Elimination method. However, if one understand the concept of substitution to reduce the number of unknowns to one, then it will simpler.

To learn more methods to solve simultaneous equations, click this here for post.
:-)

Elimination method | Simultaneous Equations

To solve simultaneous equations, the first requirement is to make sure the number of unknown variables are equal or less than the number of unique equations relating the unknowns.

By the use of elimination method, we apply the concept of removing one unknown (randomly selected) at a time. This will reduce the number of unknowns and will definitely make the solution simpler. By elimination, we try to first make any two equations having at least one unknown similar (step 1)

Example: Step 1

3x + 2y = 5 --- (A)

x + y = 2 --- (B)

We will target one unknown to be eliminated (removed) ==> Select "y" to be removed.

To make equations (B) having the "y" similar to equation (A), we need to multiply (B) by 2. Therefore new equation (B) becomes ==> 2x + 2y = 4 -----(C)

New set of equations:

3x + 2y = 5 ---(A)

2x + 2y = 4 --- (C)

After making the "y" term look the same, we move on to eliminating them (step 2).

Step 2:

Now, we see that the "2y" are the same. *** To remove the"y" part, we can now subtract equation (A) and (C). This gives us the result,

(3x - 2x) + (2y - 2y) = 5 - 4 ===> x + 0 = 1 ===> x = 1 (Answer).

Note: This removal of y through subtraction is known as ELIMINATION.

Step 3 and the final part is to replace the found answer for x back into any one of the equations (A), (B) or (C). Let's select equation (B) as it is simpler.

x + y = 2 ==> 1 + y = 2 ===> y = 2 - 1 = 1 . Therefore y = 1 (Answer).

By eliminating one unknown from the equations given, we can see that the solution becomes simpler. The first found answer is later replace back into any one of the equations to extract out the other unknowns.

To conclude NOTE:

1) Do learn the concept and method of elimination as it can be applied to any number of equations not limited to the example above which has only two equations.

2) This elimination method applies only for same order type of equations. (Order means the power of the unknown).

Example:
x^2 + y^2 = 18 ---- (A)

3x^2 + 4y^2 = 63 ----(B) ===>;Equation (A) & (B) has same order of 2.

For more information regarding solving simultaneous equations, you are invited to this post.

Happy learning maths. Think simple, and maths will be simple!