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Simplifying a(2b + -3c) + -2b(5c + -3a) + 3c(a + -2b) = 0 (2b * a + -3c * a) + -2b(5c + -3a) + 3c(a + -2b) = 0 (2ab + -3ac) + -2b(5c + -3a) + 3c(a + -2b) = 0 Reorder the terms: 2ab + -3ac + -2b(-3a + 5c) + 3c(a + -2b) = 0 2ab + -3ac + (-3a * -2b + 5c * -2b) + 3c(a + -2b) = 0 2ab + -3ac + (6ab + -10bc) + 3c(a + -2b) = 0 2ab + -3ac + 6ab + -10bc + (a * 3c + -2b * 3c) = 0 2ab + -3ac + 6ab + -10bc + (3ac + -6bc) = 0 Reorder the terms: 2ab + 6ab + -3ac + 3ac + -10bc + -6bc = 0 Combine like terms: 2ab + 6ab = 8ab 8ab + -3ac + 3ac + -10bc + -6bc = 0 Combine like terms: -3ac + 3ac = 0 8ab + 0 + -10bc + -6bc = 0 8ab + -10bc + -6bc = 0 Combine like terms: -10bc + -6bc = -16bc 8ab + -16bc = 0 Solving 8ab + -16bc = 0 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '16bc' to each side of the equation. 8ab + -16bc + 16bc = 0 + 16bc Combine like terms: -16bc + 16bc = 0 8ab + 0 = 0 + 16bc 8ab = 0 + 16bc Remove the zero: 8ab = 16bc Divide each side by '8b'. a = 2c Simplifying a = 2c
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