# Solve these simultaneous equations that involve vectors

• chwala
In summary, the problem is that two vectors are not linearly independent and that the solution is to find a way to make them independent.
chwala
Gold Member
Homework Statement
See attached
Relevant Equations
understanding of vectors and simultaneous equation
Find the question and solution here;

Ok, i was able to solve this by using,
##3A=3ax+12ay+6bx+3by+3b##
##2B=2ay-4ax+4a+4bx-6by-2b##

leading us to the simultaneous equation;
##7x+10y=4##
##2x+9b=-5##
##x=2## and ##y=-1##

I had initially tried the approach of using ##3A=2B## →##B=1.5A## ...Then on substituting this in ##A##, I got
##B_1= 1.5a(x+4y) + 1.5b(2x+y+1)## this ought to be equal to,
##B_2 = a(y-2x+2)+b(2x-3y-1)##

giving me the simultaneous equation,
##3.5x+5y=2##
##x+4.5b=-2.5##

aaaaargh this is also correct, i had missed out on a term...

any other approach guys welcome...

Last edited:
chwala said:
I had initially tried the approach of using ##3A=2B## →##B_1=1.5A## ...Then on substituting this in ##A##, I got
##B_1= 1.5a(x+4y) + 1.5b(2x+y+1)## this ought to be equal to,
This looks wrong. I don't understand what you did here.
If you are trying to figure out alternative methods to simultaneous equations, why? IMHO, if you really understood simultaneous equations, you would not be looking for something else.

FactChecker said:
This looks wrong. I don't understand what you did here.
If you are trying to figure out alternative methods to simultaneous equations, why? IMHO, if you really understood simultaneous equations, you would not be looking for something else.
I think both simultaneous equations are correct...I will check and confirm later...I was thinking of a possibility of a much faster approach but I guess it doesn't matter. Cheers man.

Note;
I just amended a term in the second simultaneous equation.
Take note that in the second simultaneous equation, i was equating ##B##=##B## ...

chwala said:
I think both simultaneous equations are correct...I will check and confirm later...I was thinking of a possibility of a much faster approach but I guess it doesn't matter. Cheers man.

Note;
I just amended a term in the second simultaneous equation.
Take note that in the second simultaneous equation, i was equating ##B##=##B## ...
Note that ##\mathbf a## and ##\mathbf b## are vectors. Something like:

chwala said:
##2x+9b=-5b##
Cannot be right, as ##x## is a number.

The key to the problem is that two non-colinear vectors are linearly independent.

jim mcnamara
PeroK said:
Note that ##\mathbf a## and ##\mathbf b## are vectors. Something like:Cannot be right, as ##x## is a number.

The key to the problem is that two non-colinear vectors are linearly independent.
Sorry a typo, let me amend it...the vectors are ##a## and ##b## ...##x## and ##y## are scalar quantities.

chwala said:
Sorry a typo, let me amend it...the vectors are ##a## and ##b## ...##x## and ##y## are scalar quantities.
Okay, I see what you've done now. Writing ##b## instead of ##\mathbf b## and then mistyping ##b## instead of ##y## and ##-5b## instead of ##-5## was too many errors for me to follow what you were doing.

PeroK said:
Okay, I see what you've done now. Writing ##b## instead of ##\mathbf b## and then mistyping ##b## instead of ##y## and ##-5b## instead of ##-5## was too many errors for me to follow what you were doing.
My silly me...Will try to go a bit slower in my typing...

## 1. What are simultaneous equations involving vectors?

Simultaneous equations involving vectors are equations that contain multiple variables and involve vectors as their coefficients. These equations are used to represent relationships between multiple quantities and can be solved using various methods.

## 2. How do you solve simultaneous equations involving vectors?

There are several methods for solving simultaneous equations involving vectors, such as substitution, elimination, and graphical methods. These methods involve manipulating the equations to eliminate variables and find a solution that satisfies all equations simultaneously.

## 3. Can you give an example of solving simultaneous equations involving vectors?

Sure, let's say we have the following two equations:
2x + 3y = 10
4x + 2y = 12
We can solve this system of equations by elimination, multiplying the first equation by -2 and adding it to the second equation to eliminate the x variable. This gives us y = 4. Then, we can substitute this value of y into either of the original equations to solve for x. In this case, x = 1.

## 4. What are the applications of solving simultaneous equations involving vectors?

Solving simultaneous equations involving vectors is useful in many fields, such as physics, engineering, and economics. It can be used to model real-life situations and make predictions based on the relationships between different variables.

## 5. Are there any tips for solving simultaneous equations involving vectors?

One tip is to start by identifying which method will be most efficient for solving the particular set of equations. It can also be helpful to simplify the equations by dividing both sides by a common factor or rearranging terms before using a specific method. Additionally, checking the solution by plugging it back into the original equations can help avoid errors.

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