Algebra/Algebra

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Question
11x+15y+23=0, 7x-2y-20=0

I tried,

11x+15y+23=0------1
7x-2y-20=0-----------2

11x+15y=-23
=>y=-23x-11x/15 (substituting the value of y in eq 2)

11x+15(-23x-11x/15)=20

Answer
The correct way to try that is to say
1) 11x + 15y + 23 = 0
2) 7x - 2y - 20 = 0.

Correctly solving the 1st equation for x gives ...
Subtract off 11x + 23 from both sides, giving 15y = -23 - 11x.
Divide by 15, giving y = -23/15 - 11x/15.

Put that into (2) for y and solve.  That gives 7x - 2(-23/15) -2(-11x/15) - 20 = 0.
The negatives in the 2nd terms cancel, giving 2(23/15),and
the negatives in the 3rd equation cancel, giving 2(11x/15).

Note that 2(23/15) = 46/15 and 2(11x/15) = 22x/15, so the equation
7x - 2(-23/15) -2(-11x/15) - 20 = 0 turns into 7x + 46/15 + 22x/15 - 20 = 0.

Since 7x is really 105x/15 and -20 is -300/15, we can put those in and get
105x/15 + 46/15 + 22x/15 - 300/15 = 0.

Multiplying both sides by 15 drops all the fractions and we are left with
105x + 46 + 22x - 300 = 0.

Combining like terms gives (105+22)x + 46-300 = 0.
That reduces to 127x – 254 = 0.

Adding 254 to both sides gives 127x = 254.
Dividing both sides by 127 gives x = 2.

Using equation (1), we then solve for y.  Putting in x=2 gives 22 + 15y + 23 = 0.
Since 22+23=45, we have 45+15y=0.  Subtracting 45 from both sides gives 15y=-45.
Dividing both sides by 15 gives y=-3.

Taking x=2 and y =-3, the 2nd equation can be used to check.
The left side gives 7(2) -2(-3) – 20  = 14 + 6 – 20 = 20 – 20 = 0,
and that’s what is on the right side, so this has been done correctly.  

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Scott A Wilson

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