How do you solve the simultaneous equations x2+y2=29 and yx=3?

2 Answers
Jul 19, 2015

Use the second equation to provide an expression for y in terms of x to substitute into the first equation to give a quadratic equation in x.

Explanation:

First add x to both sides of the second equation to get:

y=x+3

Then substitute this expression for y into the first equation to get:

29=x2+(x+3)2=2x2+6x+9

Subtract 29 from both ends to get:

0=2x2+6x20

Divide both sides by 2 to get:

0=x2+3x10=(x+5)(x2)

So x=2 or x=5

If x=2 then y=x+3=5.

If x=5 then y=x+3=2

So the two solutions (x,y) are (2,5) and (5,2)

Jul 19, 2015

(x=5andy=2)or(x=2andy=5)

Explanation:

Since you have both x2+y2=29 and yx=3,

You want to combine these two equations into one equation with a single variable, solve it and then solve for the other variable. An example on how to do this goes like this:

yx=3y=x+3 and we have y2=x2+6x+9

Since x2+y2=29, substitute the expression for y2 into this:

2x2+6x+9=29, so 2x2+6x20=0.

We can solve for x using the quadratic formula:
x=6±3642(20)22=34±14196=6±144
So x=5 or x=2.

Since y=x+3, this gives (x=5andy=2)or(x=2andy=5).