Can This Method Solve the Modified System of Linear Equations?

In summary, we have a system of equations with a unique solution, and we need to find the set of solutions for a similar system. Using Cramer's rule, we can find that the solution for the first equation leads to a linear combination of vectors a1 and a2, which indicates that the vectors a1, a2, and a3 are linearly independent. This allows us to rearrange the equations and solve for (x, y, z) using the fact that a1, a2, and a3 are linearly independent. However, if Cramer's rule is not allowed, another approach could be to look at the corresponding vector equation and use the same fact of linear independence to solve for the variables.
  • #1
MathematicalPhysicist
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we have the system a_i1x+a_i2y+a_i3z=b_i i=1,2,3
and we are given that it has a unique solution (1,2,0)
i need to find the set of solutions for the next system:
a_i1x+b_iy+a_i3z=a_i2
what i did is:
we have the solution for the first equation then we have a_i1+2a_i2=b_i
so we put it in the second equation and we have:
a_i1x+(a_i1+2a_i2)y+a_i3z=a_i2
(a_i1x+a_i2y+a_i3z)+(a_i1+a_i2)y=a_i2
b_i+(a_i1+a_i2)y=a_i2
(a_i1+2a_i2)+(a_i1+a_i2)y=a_i2
from here i found the solution, is this method valid?
 
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  • #2
Hint:
Look up Cramer's rule, and see what you can find out with that.
 
  • #3
do you know perhaps another method to solve this other than cramer's rule?
cause I am not supposed to use it here.
 
  • #4
Try to look at the corresponding vector equation: [tex]\vec{a}_{1}x + \vec{a}_{2}y + \vec{a}_{3}z = \vec{b}[/tex].
 
  • #5
i don't understand, radou can you elaborate?
 
  • #6
loop quantum gravity said:
i don't understand, radou can you elaborate?

Well, after plugging the solution (x, y, z) = (1, 2, 0) into the vector equation, you find out that the vector b can be written as a linear combination of vectors a1 and a2, right? So, that means that the vector b lies in the same plane as a1 and a2 do. Hence, you know that the vectors a1, a2 and a3 are linearly independent. Now, after plugging in the linear combination a1 + 2a2 = b into the equation a1x + by + a3z = a2, and rearranging, you can use the fact that a1, a2 and a3 are linearly independent to solve the equation.
 
  • #7
when you mean rearranging, you mean like i did i.e:
a1x+a2y+a3z+(a1+a2)y=a2
but how do i find for (x,y,z) with your approach?
i mean if they are independent then any combination of them, its coeffiencts are zero like this a1x+a2y+a3z=0 then x=y=z=0, but how do i use it here?
 
  • #8
I meant a1x + (a1+2a2)y+a3z = a2 =>
a1(x+1) + a2(2y-1)+a3z = 0. Now use the fact that a1, a2 and a3 are linearly independent.

Edit: too bad you're not allowed to use Cramer's rule, as Arildno suggested, since this problem seems to be ideal for its application.
 
  • #9
ok, thanks i understand now.
 

Related to Can This Method Solve the Modified System of Linear Equations?

1. What is a linear equation?

A linear equation is a mathematical statement that describes a relationship between two variables using a straight line. It is typically written in the form of y = mx + b, where m represents the slope of the line and b represents the y-intercept.

2. How do I solve a linear equation?

To solve a linear equation, you need to isolate the variable on one side of the equation by using inverse operations. For example, if the equation is 2x + 3 = 9, you would subtract 3 from both sides to get 2x = 6, and then divide both sides by 2 to get x = 3.

3. Can a linear equation have more than one solution?

Yes, a linear equation can have infinitely many solutions. This is because a linear equation represents a straight line, and any point on that line is a solution to the equation.

4. What is the difference between a linear equation and a linear function?

A linear equation is a mathematical statement, while a linear function is a mathematical relationship between two quantities where the output is directly proportional to the input. In other words, a linear function is a type of linear equation that represents a straight line on a graph.

5. How are linear equations used in real life?

Linear equations are used in many real-life situations, such as determining the cost of a product based on the number of items purchased, calculating the distance traveled given a constant speed, and predicting future values based on past trends. They are also commonly used in fields such as economics, engineering, and physics.

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