SOLUTION: How do you set up the equation here? A recent basic model of a particular automobile had a price of $14,685. The basic model with the added features of automatic transmission and

Question 1134432: How do you set up the equation here? A recent basic model of a particular automobile had a price of $14,685. The basic model with the added features of automatic transmission and power locks was $16,070. The basic model with air conditioning and power locks was $15,580. The basic model with AC and automatic transmission was $15,925. What was the individual cost of each of the three options?
i have:
Basic = $14,685
B+T+L= $14,685 + T + L = $16070
B+A+L= $14,685 + A + L = $15,580
B+A+T = $14,685 +A+T = $15,925
What is the next step?
Found 4 solutions by greenestamps, josgarithmetic, ikleyn, MathTherapy:
Answer by greenestamps(13198)   (Show Source): You can put this solution on YOUR website!

From your first equation,

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Similarly, from your other two equations, you can get values for A+L and A+T.

Then the trick is to add the three resulting equations; that will give you the value of 2A+2L+2T. Divide by 2 to find the value of A+L+T. Then compare that value to each of the equations you have for the sum of two of the variables to find the value of each individual variable.

Probably sounds confusing if you haven't done it before; here's an example.
A+B = 34 A+C = 53 B+C = 61 ---------------- 2A+2B+2C = 148 A+B+C = 148/2 = 74 A = (A+B+C)-(B+C) = 74-61 = 13 B = (A+B+C)-(A+C) = 74-53 = 21 C = (A+B+C)-(A+B) = 74-34 = 40

Answer by josgarithmetic(39616)   (Show Source): You can put this solution on YOUR website!
b, Basic Model
t, Automatic Transmission
L, power Locks
a, Air conditioning

b=14685

b can be subtracted on both sides of each equation.
You are given value for b.

.
.
.
Answer by ikleyn(52776)   (Show Source): You can put this solution on YOUR website!
.

Let start with the system from your post B = 14685 (1) 14685 + T + L = 16070 (2) 14685 + A + L = 15580 (3) 14685 + A + T = 15925 (4) We have 4 independent linear equations in 4 unknown, so the solution does exist, for sure. The system has very special structure, and I will show you VERY SPECIAL trick to easy solve it, which works PERFECTLY for this structure. First, subtract the value 14685 from both sides of each equation (2), (3) and (4). You will get T + L = 16070 - 14685 = 1385 (5) A + L = 15580 - 14685 = 895 (6) A + T = 15925 - 14685 = 1240 (7) The trick starts from this point. First, add all three the equations (5), (6) and (7). You will get 2T + 2A + 2L = 1385 + 895 + 1240 = 3520, or, dividing by 2 both sides, T + A + L = 1760 (8) Now subtract equation (5) from equation (8). You will get A = 1760 - 1385 = 375. Next, subtract equation (6) from equation (8). You will get T = 1760 - 895 = 865. Finally, subtract equation (7) from equation (8). You will get L = 1710 - 1240 = 520. The problem is just solved. The ANSWER is $375 for A; $865 for T, and $520 for L.

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If you want to see other similar solved problems, look into my lesson
    - The tricks to solve some word problems with three and more unknowns using mental math
in this site.

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Answer by MathTherapy(10551)   (Show Source): You can put this solution on YOUR website!
How do you set up the equation here? A recent basic model of a particular automobile had a price of $14,685. The basic model with the added features of automatic transmission and power locks was $16,070. The basic model with air conditioning and power locks was $15,580. The basic model with AC and automatic transmission was $15,925. What was the individual cost of each of the three options?
i have:
Basic = $14,685
B+T+L= $14,685 + T + L = $16070
B+A+L= $14,685 + A + L = $15,580
B+A+T = $14,685 +A+T = $15,925
What is the next step?

Using your equations:
14,685 + T + L = 16,070_____T + L = 16,070 - 14,685_____T + L = 1,385 ----- eq (i)
14,685 + A + L = 15,580_____A + L = 15,580 - 14,685_____A + L = 895 ----- eq (ii)
14,685 + A + T = 15,925_____A + T = 15,925 - 14,685_____A + T = 1,240 ----- eq (iii)
Subtract eq (i) from eq (ii) to get:....................A - T = - 490 ----- eq (iv)
Add eqs (iv) & (iii) to get:...........................2A = 750
A, or cost of automatic transmission =
I'm sure you'll be able to find the other costs.
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