linear equation

Question

4a+6b=1.56 & 9a+7b=2.60, a = apple & b = banana. what is the cost of one apple and one banana? seeking the simplest and fastest way preferably without using a calculator.

this is a timed question, so i plan to re-use the method taught to me.

Answer 1

4a + 6b = 1.56

9a + 7b = 2.60    Add these two equations together vertically!

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13a + 13b = 4.16    Now divide this whole equation by 13 to see what a + b would be

 

(13a)/13 + (13b)/13 = (4.16)/13

a + b = 4.16/13

 

Do some long division if you don't want to use a calculator.

13 goes into 41 3 times with a 2 remainder. Bring the 6 down next to the 2, 13 goes into 26 2 times exactly.

 

so a + b = 0.36. One apple and one banana cost a total of 36 cents.

 

 

Answer 2

The easiest way (without a calculator) to solve this system of equations is by "elimination".  

 

Multply the first equation by (-9).

Multiply the second equation by (4).

 

This will make the coefficient of "a" a (+36) on top

and the coefficient of "a" a (-36) on the bottom.

 

Then add vertically.  "a" will be eliminated, can then solve for "b".

 

(-9)[4a+6b]=(-9)(1.56)

(4)[9a+7b]=(4)(2.60)

 

Now we have

 

 -36a - 54b  =-14.04

+36a + 28b = 10.40

 

Add "a"s with "a"s and "b"s with "b"s and constants with constants (add vertically)

 

And get

0a - 26b = -3.64

 

Divide both sides by -26 and get

 

b=$0.14  (or bananas cost 14 cents each)

 

Now plug this "b" back into one of the ORIGINAL equations to find "a"

 

4a + 6(0.14)=1.56

4a + 0.84 = 1.56

 

Subtract 0.84 from both sides.

4a = 0.72

 

Divide both sides by 4

 

a = $0.18  (or apples cost 18 cents each).

 

 

Now check by plugging both values into the second equation ( the one we did NOT USE to find a)

 

Get 9(.18)+7(.14)=1.62+0.98=2.60  :-)  It works!!!  so we have found the solution.

 

Final answer:  Apples cost 18 cents each,  bananas cost 14 cents each.