Tell whether the linear system has ONE SOLUTION, NO SOLUTION, or INFINITELY MANY SOLUTIONS.
4x-y=2
-x+3y=9
since the lines represented by the two equations are not parallel, they must intersect. There is one solution.
To determine whether the linear system has one solution, no solution, or infinitely many solutions, we can use the method of solving the system known as Gaussian elimination.
We start by writing the system of equations in augmented matrix form:
[4 -1 | 2]
[-1 3 | 9]
First, we want to eliminate the -1 coefficient in the second equation by multiplying the first equation by 1/4:
[ 4 -1 | 2]
[ -1 3 | 9]
Next, we perform the row operation R2 = R2 + R1/4:
[ 4 -1 | 2]
[ 0 2.75 | 11/4]
To get rid of the decimal in the second row, we can multiply the entire row by 4 to clear the fractions:
[ 4 -1 | 2]
[ 0 11 | 11 ]
Now we divide the second equation by 11 to simplify it further:
[ 4 -1 | 2]
[ 0 1 | 1 ]
Finally, we eliminate the -1 coefficient in the first equation by multiplying the second equation by -1 and adding it to the first equation:
[ 4 0 | 1 ]
[ 0 1 | 1 ]
The resulting system is:
4x = 1
y = 1
We can solve for x and y:
From the first equation, we have: x = 1/4.
From the second equation, we have: y = 1.
Therefore, the linear system has one unique solution, where x = 1/4 and y = 1.