How do you solve x^2+y^2=109 and x-y= - 7 using substitution?

4 Answers
Mar 28, 2018

:."The Solution Set="{(x,y)=(3,10),(-10,-3)}.

Explanation:

x-y=-7 rArr x=y-7.

Sub.ing in, x^2+y^2=109, we get,

(y-7)^2+y^2=109, or,

2y^2-14y+49-109=0, i.e.,

y^2-7y-30=0.

Using 10xx3=30, and, 10-3=7, we get,

ul(y^2-10y)+ul(3y-30)=0.

:. y(y-10)+3(y-10)=0.

:. (y-10)(y+3)=0.

:. y=10, or, y=-3.

y=10, &, x=y-7 rArr x=3,and, y=-3 rArr x=-10.

These roots satisfy the given eqns.

:."The Solution Set="{(x,y)=(3,10),(-10,-3)}.

Mar 28, 2018

x = - 10 , y = -3 and x = 3, y = 10

Explanation:

Rearrange x - y = -7 so x = y - 7
Now substitute "y - 7 " for the x in the first equation.

x^2 + y^2 = 109 becomes (y - 7)^2 + y^2 = 109
Expand the bracket:
y^2 - 7y - 7y +49 + y^2 = 109

Collect like terms:
2y^2 - 14y + 49 = 109

subtract 109 from both sides:
2y^2 - 14y - 60 = 0

Factorise:
(2y + 6)(y - 10) = 0

so 2y + 6 = 0 or y - 10 = 0

then y = -3 or y = 10

Now put these two values into x - y = - 7
when y = - 3,
x - (- 3) =- 7
x + 3 = - 7
x = - 10

when y = 10,
x -10 = - 7
x = 3

So the answers are:
x = - 10 and y = - 3
and
x = 3 and y = 10

Mar 28, 2018

Express the implicit relation in terms of one variable only. See method below.

Explanation:

You have two equations:

x^2 + y^2 = 109

x-y=-7

You can express x in terms of y or express y in terms of x. Either way is fine.

Since x-y=-7, that means y=x+7

Using this substitution we can write x^2 + (x+7)^2 = 109

This will give us a quadratic expression where we can solve for x.

x^2 + (x+7)^2 = x^2 + (x^2 + 14x + 49) = 109

Simplifying this we get:
x^2 + 7x -30 = 0

This can be factorized to (x-3)(x+10)=0 giving us the solutions x = 3 and x=-10.

We can then solve for y using the equation y=x+7

If x=3 then y=10

If x=-10 then y=-3

Mar 28, 2018

(x,y)=(3,10)color(white)("xxx")"or"color(white)("xxx")(x,y)=(-10,-3)

Explanation:

Given
[1]color(white)("XXX")x^2+y^2=109
[2]color(white)("XXX")x-y=-7

Note that [2] implies
[3]color(white)("XXX")x=y-7

Substituting (y-7) for x in [1]
[4]color(white)("XXX")(y-7)^2+y^2=109

Expanding and simplifying the left side of [4]
[5]color(white)("XXX")2y^2-14y+49=109

Converting to standard polynomial form by subtracting 109 from both sides
[6]color(white)("XXX")2y^2-14y-60=0

Factoring
[7]color(white)("XXX")2(y-10)(y+3)=0

Which implies:
{: ([8a]color(white)("XX")y=10,color(white)("XX")"or"color(white)("XX"), [8b]color(white)("XX")y=-3), ("Substituting "10" for "x,,"Substituting "-3" for "x), ("in [3]",,"in [3]"), ([9a]color(white)("XX")x=10-7=3,,[9b]color(white)("XX")x=-3-7=-10) :}