Question

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.

Answer 1

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