This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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You're on a roll — To determine if a pair of linear equations is consistent or inconsistent, we compare the ratios of their coefficients. For a system of equations: a_1x + b_1y + c_1 = 0 a_2x + b_2y + c_2 = 0 • If (a_1)/(a_2) ≠ (b_1)/(b_2), the system is consistent (unique solution). • If (a_1)/(a_2) = (b_1)/(b_2) = (c_1)/(c_2), the system is consistent (infinitely many solutions). • If (a_1)/(a_2) = (b_1)/(b_2) ≠ (c_1)/(c_2), the system is inconsistent (no solution). Let's solve the visible parts of the problem. Part (i): The given equations are: 3x + 2y = 5 3x + 2y - 5 = 0 2x - 3y = 7 2x - 3y - 7 = 0 Step 1: Identify the coefficients. Here, a_1 = 3, b_1 = 2, c_1 = -5 And a_2 = 2, b_2 = -3, c_2 = -7 Step 2: Calculate the ratios. (a_1)/(a_2) = (3)/(2) (b_1)/(b_2) = (2)/(-3) = -(2)/(3) (c_1)/(c_2) = (-5)/(-7) = (5)/(7) Step 3: Compare the ratios. Since (3)/(2) ≠ -(2)/(3), we have (a_1)/(a_2) ≠ (b_1)/(b_2). Step 4: Conclude consistency. Therefore, the pair of equations is consistent. Part (iii): The given equations are: (3)/(2)x + (5)/(3)y = 7 (3)/(2)x + (5)/(3)y - 7 = 0 9x - 10y = 14 9x - 10y - 14 = 0 Step 1: Identify the coefficients. Here, a_1 = (3)/(2), b_1 = (5)/(3), c_1 = -7 And a_2 = 9, b_2 = -10, c_2 = -14 Step 2: Calculate the ratios. (a_1)/(a_2) = (3)/(2)9 = (3)/(2 × 9) = (3)/(18) = (1)/(6) (b_1)/(b_2) = (5)/(3)-10 = (5)/(3 × (-10)) = (5)/(-30) = -(1)/(6) (c_1)/(c_2) = (-7)/(-14) = (1)/(2) Step 3: Compare the ratios. Since (1)/(6) ≠ -(1)/(6), we have (a_1)/(a_2) ≠ (b_1)/(b_2). Step 4: Conclude consistency. Therefore, the pair of equations is consistent. Part (v): The given equations are: -x + 2y = 8 -x + 2y - 8 = 0 2x + 3y = 12 2x + 3y - 12 = 0 Step 1: Identify the coefficients. Here, a_1 = -1, b_1 = 2, c_1 = -8 And a_2 = 2, b_2 = 3, c_2 = -12 Step 2: Calculate the ratios. (a_1)/(a_2) = (-1)/(2) (b_1)/(b_2) = (2)/(3) (c_1)/(c_2) = (-8)/(-12) = (2)/(3) Step 3: Compare the ratios. Since -(1)/(2) ≠ (2)/(3), we have (a_1)/(a_2) ≠ (b_1)/(b_2). Step 4: Conclude consistency. Therefore, the pair of equations is consistent. Send me the next one 📸