determine whether the system of alinear equqtion has one and only one solutions or no solutions find all solutions whenever they exits
3x- 4y =12
6x -8y =24
You will notice that by multiplying the fist equation by two (2), you get exactly the second equation.
This means that the second equation is a linear transformation of the first, and geometrically means that the lines corresponding to the two equations are coincident.
Algebraically, it means that there are infinite solutions, depending on an arbitrary finite value of one of the variables.
One possible solution set is:
Let x=t (any finite real value)
then
(x,y)=(t,(3t-12)/4)
To determine whether the system of linear equations has one and only one solution, no solution, or infinitely many solutions, we need to solve the system of equations and analyze the resulting system.
The given system of equations is:
1) 3x - 4y = 12
2) 6x - 8y = 24
To solve for x and y, we can use the method of substitution or elimination. Let's use the elimination method to solve these equations:
Multiply equation 1) by -2 to match the coefficients of x with equation 2):
-6x + 8y = -24 (equation 3)
Now, let's add equation 2) and equation 3) together:
6x - 8y + (-6x + 8y) = 24 + (-24)
0 = 0
When we simplify, we get the equation 0 = 0. This equation holds true for all values of x and y.
Explanation: When we obtained the equation 0 = 0, it implies that the two equations are equivalent or represent the same line. This situation means that the system of equations has infinitely many solutions. In other words, any values of x and y that satisfy one equation will also satisfy the other equation.
Therefore, the system of linear equations has infinitely many solutions.