One week, the music store sold 2 trumpets, 3 clarinets, and 5 violins for $1240. the next week, they sold 3 trumpets, 1 clarinet, and 4 violins for $1027. the following week, they sold 5 trumpets, 7 clarinets, and 2 violins for $2091. find the cost of a trumpet, a clarinet would, and a violin.

Trumpet = $174 Clarinet = $149 Violin = $89 Let's write some equations to express what we know. 2t + 3c + 5v = 1240 3t + 1c + 4v = 1027 5t + 7c + 2v = 2091 So we have 3 unknowns and 3 equations. The 2t and 3t are rather interesting in the 1st 2 equations since the 3rd equation has 5t. So if we subtract the 1st and 2nd equations, we can cancel out the t values. So 5t + 7c + 2v = 2091 - (3t + 1c + 4v = 1027) = 2t + 6c - 2v = 1064 - (2t + 3c + 5v = 1240) = 3c - 7v = - 176 Now we can express c in terms of v. 3c - 7v = - 176 3c = 7v - 176 c = (7/3) v - 58 2/3 Substitute the expression (7/3) v - 58 2/3 for c in the expression 2t + 3c + 5v = 1240 giving 2t + 3 ((7/3) v - 58 2/3) + 5v = 1240 and solve for t, first distribute the 3 2t + 7v - 176 + 5v = 1240 Merge the v terms 2t + 12v - 176 = 1240 Add 176 to both sides 2t + 12v = 1240 + 176 = 1416 Subtract 12 v from both sides 2t = 1416 - 12v Divide both sides by 2 t = 708 - 6v Now substitute both 708 - 6v for t and (7/3) v - 58 2/3 for c in 3t + 1c + 4v = 1027 and solve for v 3t + 1c + 4v = 1027 3 (708 - 6v) + (7/3) v - 58 2/3 + 4v = 1027 Distribute the 3 3 (708 - 6v) + (7/3) v - 58 2/3 + 4v = 1027 2124 - 18v + (7/3) v - 58 2/3 + 4v = 1027 Combine terms 2065 1/3 - (11 2/3) v = 1027 Subtract 2065 1/3 from both sides - (11 2/3) v = 1027 - 2065 1/3 = - 1038 1/3 Divide both sides by - 11 2/3 v = 89 So we know the violins cost $89 each. Plugging that value into the formula t = 708 - 6v which we created earlier t = 708 - 6v t = 708 - 6 * 89 = 708 - 534 = 174 So we know that a trumpet costs $174 Now plug the price of a violin into the formula c = (7/3) v - 58 2/3 to get the cost of a clarinet. c = (7/3) v - 58 2/3 c = (7/3) 89 - 58 2/3 = 207 2/3 - 58 2/3 = 149 And a clarinet is $149 Let's verify 2t + 3c + 5v = 2*174 + 3*149 + 5*89 = 348 + 447 + 445 = 1240 3t + 1c + 4v = 3*174 + 1*149 + 4*89 = 522 + 149 + 356 = 1027 5t + 7c + 2v = 5*174 + 7*149 + 2*89 = 870 + 1043 + 178 = 2091 All the expressions match what was given, so the prices are Trumpet = $174 Clarinet = $149 Violin = $89