Solving Linear Systems of Equations

12 apples & 8 guavas total cost is Rs. 76/-. 8 apples & 12 guavas cost is Rs. 64/-. What is the cost of each apple & guava?

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Nov 18, 2010
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Solving Linear Systems of Equations
by: Karin

I'm not sure what you mean by: Rs. 76/- I'm thinking that this is the form of currency that you use in your country.

If 12 apples and 8 guavas cost RS. 76, then we could write this equation as:

Let a = apples Let g = guavas

12a + 8g = 76


If 8 apples and 12 guavas cost RS. 64, then we could write this equation as:

8a + 12g = 64

Now that you have written two equations, you have written a system of equations. With this system you can choose one of three methods to solve: substitution, linear combinations, or graphing.

Since the equations are written in standard form, I find the easiest method to use is linear combinations.

Step 1: We must create one set of opposite terms.

I am going to create opposite "g" terms by multiplying the first equation by -3 and the second equation by 2.

-3[12a + 8g] = [76]-3

-36a - 24g = -228

Now the second equation we must multiply by 2:

2[8a +12g] = 64[2]

16a + 24g = 128

Now our two new equations are:
-36a - 24g = -228
16a + 24g = 128

Notice that we have opposite terms (-24g & 24g)

Step 2: Add the two equations:
-36a - 24g = -228
16a + 24g = 128
---------------------
-20a + 0 = -100

Step 3: Solve for a
-20a/-20 = -100/-20
a = 5

So now we know that we have 5 apples.

Step 4: Substitute 5 for a and solve for g in one of the equations.

8a + 12g = 64
8(5) + 12g = 64
40 + 12g = 64
40 -40 +12g = 64 - 40
12g = 24
12g/12 = 24/12
g = 2

So, now we know that we have 2 guavas.

The solution is 5 apples and 2 guavas.

You can substitute these numbers into the original equations to check.

Hope this helps!
Karin





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