How many solutions does the system of equations below have?
6x − 8y = 7
–10x + 9y = –2
Since the 2 lines have different slopes, obviously there is only one solution.
you were wrong
infinetly many soulutions
To determine the number of solutions for a system of equations, we can use the concept of linear independence.
First, let's write the system of equations in matrix form:
| 6 -8 | | x | | 7 |
|-10 9 | * | y | = |-2 |
Now, let's find the determinant of the coefficient matrix. The determinant tells us whether the system has unique solutions, no solution, or infinitely many solutions.
To find the determinant, we calculate:
det(A) = (6 * 9) - (-8 * -10) = 54 - 80 = -26
If the determinant is nonzero, then the system has a unique solution. If the determinant is zero, then the system has either no solution or infinitely many solutions.
Since the determinant (-26) is nonzero, we can conclude that the system of equations has a unique solution. Therefore, the system has one solution.