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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To find the value of k for which the system of equations has an infinite number of solutions, we use the condition that the ratios of the coefficients must be equal. The given system of equations is: 1) kx + 3y = (2k+1) 2) 2(k+1)x + 9y = (7k+1) For a system of linear equations a_1x + b_1y = c_1 and a_2x + b_2y = c_2 to have an infinite number of solutions, the following condition must be met: (a_1)/(a_2) = (b_1)/(b_2) = (c_1)/(c_2) Step 1: Identify the coefficients from the given equations. From equation (1): a_1 = k, b_1 = 3, c_1 = 2k+1 From equation (2): a_2 = 2(k+1), b_2 = 9, c_2 = 7k+1 Step 2: Set up the ratios according to the condition for infinite solutions. (k)/(2(k+1)) = (3)/(9) = (2k+1)/(7k+1) Step 3: Solve the first part of the equality to find k. (k)/(2(k+1)) = (3)/(9) Simplify the right side: (k)/(2(k+1)) = (1)/(3) Cross-multiply: 3k = 2(k+1) 3k = 2k + 2 Subtract 2k from both sides: 3k - 2k = 2 k = 2 Step 4: Verify the value of k with the second part of the equality. Substitute k=2 into the ratio (3)/(9) = (2k+1)/(7k+1): Left side: (3)/(9) = (1)/(3) Right side: (2(2)+1)/(7(2)+1) = (4+1)/(14+1) = (5)/(15) = (1)/(3) Since (1)/(3) = (1)/(3), the value k=2 satisfies all conditions for the system to have an infinite number of solutions. The value of k is 2. Send me the next one 📸