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To determine the nature of the system of equations given by [tex]\(3x - 6y = 20\)[/tex] and [tex]\(2x - 4y = 3\)[/tex], we follow a step-by-step process involving the calculation of the determinant and the examination of the equations:
1. Identify the coefficients:
- For the first equation [tex]\(3x - 6y = 20\)[/tex], the coefficients are [tex]\(a_1 = 3\)[/tex], [tex]\(b_1 = -6\)[/tex], [tex]\(c_1 = 20\)[/tex].
- For the second equation [tex]\(2x - 4y = 3\)[/tex], the coefficients are [tex]\(a_2 = 2\)[/tex], [tex]\(b_2 = -4\)[/tex], [tex]\(c_2 = 3\)[/tex].
2. Calculate the determinant of the coefficient matrix:
[tex]\[ \text{Determinant} = a_1 \cdot b_2 - a_2 \cdot b_1 \][/tex]
Substituting the coefficients:
[tex]\[ \text{Determinant} = 3 \cdot (-4) - 2 \cdot (-6) = -12 + 12 = 0 \][/tex]
3. Analyze the determinant:
- If the determinant is [tex]\(0\)[/tex], the system of equations might be either dependent or inconsistent. We need further examination to determine which case it is.
- If the determinant were non-zero, the system would be independent (which is not the case here).
4. Check the nature of the equations:
Since the determinant is [tex]\(0\)[/tex], we proceed by checking if the equations have a proportional relationship with their constant terms. For the system to be inconsistent, the ratios of the coefficients must not match.
Compare the ratios of corresponding constants and coefficients:
- First, compare [tex]\( \frac{a_1}{a_2} \)[/tex] and [tex]\( \frac{b_1}{b_2} \)[/tex]:
[tex]\[ \frac{a_1}{a_2} = \frac{3}{2}, \quad \frac{b_1}{b_2} = \frac{-6}{-4} = \frac{3}{2} \][/tex]
- Both ratios [tex]\(\frac{a_1}{a_2}\)[/tex] and [tex]\(\frac{b_1}{b_2}\)[/tex] are equal ([tex]\(\frac{3}{2}\)[/tex]).
- Next, compare [tex]\( \frac{a_1}{a_2} \)[/tex] and [tex]\( \frac{c_1}{c_2} \)[/tex]:
[tex]\[ \frac{c_1}{c_2} = \frac{20}{3} \][/tex]
- Here [tex]\( \frac{a_1}{a_2} = \frac{3}{2} \)[/tex] does not equal [tex]\( \frac{c_1}{c_2} = \frac{20}{3} \)[/tex].
5. Conclusion:
Since the ratios of corresponding coefficients are equal, but the ratio of [tex]\( \frac{a_1}{a_2} \)[/tex] does not equal the ratio [tex]\( \frac{c_1}{c_2} \)[/tex], this indicates that the system of equations is inconsistent.
Thus, the system of equations [tex]\(3x - 6y = 20\)[/tex] and [tex]\(2x - 4y = 3\)[/tex] is inconsistent.
Therefore, the correct answer is:
B. inconsistent.
1. Identify the coefficients:
- For the first equation [tex]\(3x - 6y = 20\)[/tex], the coefficients are [tex]\(a_1 = 3\)[/tex], [tex]\(b_1 = -6\)[/tex], [tex]\(c_1 = 20\)[/tex].
- For the second equation [tex]\(2x - 4y = 3\)[/tex], the coefficients are [tex]\(a_2 = 2\)[/tex], [tex]\(b_2 = -4\)[/tex], [tex]\(c_2 = 3\)[/tex].
2. Calculate the determinant of the coefficient matrix:
[tex]\[ \text{Determinant} = a_1 \cdot b_2 - a_2 \cdot b_1 \][/tex]
Substituting the coefficients:
[tex]\[ \text{Determinant} = 3 \cdot (-4) - 2 \cdot (-6) = -12 + 12 = 0 \][/tex]
3. Analyze the determinant:
- If the determinant is [tex]\(0\)[/tex], the system of equations might be either dependent or inconsistent. We need further examination to determine which case it is.
- If the determinant were non-zero, the system would be independent (which is not the case here).
4. Check the nature of the equations:
Since the determinant is [tex]\(0\)[/tex], we proceed by checking if the equations have a proportional relationship with their constant terms. For the system to be inconsistent, the ratios of the coefficients must not match.
Compare the ratios of corresponding constants and coefficients:
- First, compare [tex]\( \frac{a_1}{a_2} \)[/tex] and [tex]\( \frac{b_1}{b_2} \)[/tex]:
[tex]\[ \frac{a_1}{a_2} = \frac{3}{2}, \quad \frac{b_1}{b_2} = \frac{-6}{-4} = \frac{3}{2} \][/tex]
- Both ratios [tex]\(\frac{a_1}{a_2}\)[/tex] and [tex]\(\frac{b_1}{b_2}\)[/tex] are equal ([tex]\(\frac{3}{2}\)[/tex]).
- Next, compare [tex]\( \frac{a_1}{a_2} \)[/tex] and [tex]\( \frac{c_1}{c_2} \)[/tex]:
[tex]\[ \frac{c_1}{c_2} = \frac{20}{3} \][/tex]
- Here [tex]\( \frac{a_1}{a_2} = \frac{3}{2} \)[/tex] does not equal [tex]\( \frac{c_1}{c_2} = \frac{20}{3} \)[/tex].
5. Conclusion:
Since the ratios of corresponding coefficients are equal, but the ratio of [tex]\( \frac{a_1}{a_2} \)[/tex] does not equal the ratio [tex]\( \frac{c_1}{c_2} \)[/tex], this indicates that the system of equations is inconsistent.
Thus, the system of equations [tex]\(3x - 6y = 20\)[/tex] and [tex]\(2x - 4y = 3\)[/tex] is inconsistent.
Therefore, the correct answer is:
B. inconsistent.
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