Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution.
2, x, plus, 5, y, equals, minus, 7
2x+5y=
−7
x, plus, 10, y, equals, minus, 13
x+10y=
−13
To determine the solutions of the system of equations, we can rewrite the equations in a standard form.
The given equations are:
2x + 5y = -7 ...(Equation 1)
x + 10y = -13 ...(Equation 2)
To determine the number of solutions, we can put the equations in a matrix form:
[ 2 5 | -7 ]
[ 1 10 | -13 ]
Next, we can perform row reduction to determine the number of solutions:
[ 2 5 | -7 ]
[ 1 10 | -13 ]
R2 = 2R2 - R1
[ 2 5 | -7 ]
[ 0 15 | -27 ]
Both equations have the same y-coefficient, indicating that it is possible to eliminate y by subtracting a multiple of one equation from the other. If the corresponding x-coefficients were also the same, we could eliminate both x and y, resulting in no unique solutions. However, in this case, the x-coefficients are different, which means that there is a unique solution.
Therefore, the system of equations has exactly one solution.