Algebra Know-It-ALL

Now we can substitute (4x+ 2) for y in the first original equation, obtaining

3 x−π(4x+ 2) =− 1

When we apply the distributive law on the left side of the equals sign, we get

3 x− (4πx+ 2 π)=− 1

This is the equivalent of

3 x+ [−1(4πx+ 2 π)]=− 1

which simplifies to

3 x− 4 πx− 2 π=− 1

When we add 2π to each side, we get

3 x− 4 πx=− 1 + 2 π

We can use the distributive law “backward” to morph the left side of this equation, obtaining

(3− 4 π)x=− 1 + 2 π

Now we can divide through by (3 − 4 π) to get

x= (− 1 + 2 π)/(3− 4 π)

It’s a mess, all right! But we’ve found a real number that’s equal to x. We can plug this number into the SI
equation we derived earlier, getting

y= 4[(− 1 + 2 π)/(3− 4 π)]+ 2

= (− 4 + 8 π)/(3− 4 π)+ 2

Believe it or not, this can be simplified. But we must take a step back, and then we can take two steps for-
ward. Let’s “complexify” the number 2 and write it as twice the denominator in the fraction above, divided
by that denominator. The idea is to get a common denominator, add some fractions, and get a simpler
expression as a result. In mathematical terms,

2 = 2(3 − 4 π)/(3− 4 π)

It takes some intuition to see, in advance, how a scheme like this will work. (With practice, you’ll develop
this “sixth sense.”) Our solution for y can now be rewritten as

y= (− 4 + 8 π)/(3− 4 π)+ 2(3 − 4 π)/(3− 4 π)

This gives us a sum of two fractions with the common denominator (3 − 4 π). Therefore:

y= [(− 4 + 8 π)+ 2(3 − 4 π)]/(3− 4 π)

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