To determine the nature of the solutions for the system of linear equations given by:
1. [tex]\( 3x - 5y = 7 \)[/tex]
2. [tex]\( -6x + 10y = 7 \)[/tex]
We will follow a systematic approach to analyze the system.
First, we note the coefficients of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in each equation:
1. [tex]\( a_1 = 3 \)[/tex], [tex]\( b_1 = -5 \)[/tex], [tex]\( c_1 = 7 \)[/tex]
2. [tex]\( a_2 = -6 \)[/tex], [tex]\( b_2 = 10 \)[/tex], [tex]\( c_2 = 7 \)[/tex]
We calculate the determinant of the coefficient matrix:
[tex]\[ D = a_1 \cdot b_2 - a_2 \cdot b_1 \][/tex]
[tex]\[ D = 3 \cdot 10 - (-6) \cdot (-5) \][/tex]
[tex]\[ D = 30 - 30 \][/tex]
[tex]\[ D = 0 \][/tex]
Since the determinant [tex]\( D \)[/tex] is zero, we know that the system of equations is either dependent (infinitely many solutions) or inconsistent (no solutions). To determine which case applies, we compare the ratios of the coefficients and constants:
We check:
[tex]\[ \frac{a_1}{a_2} = \frac{3}{-6} = -\frac{1}{2} \][/tex]
[tex]\[ \frac{b_1}{b_2} = \frac{-5}{10} = -\frac{1}{2} \][/tex]
[tex]\[ \frac{c_1}{c_2} = \frac{7}{7} = 1 \][/tex]
We see that:
[tex]\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \][/tex]
This indicates that the two lines are parallel but not coincident. Therefore, the equations have no solution.
Thus, the correct answer is:
C) No solution
