Question

Solve this system of equations and provide a graphical representation of the solution. (12)

x² + y² = 5

x + y = 1

Answer 1

We can solve this system by substitution.

From the second equation we have:

$$y=1-x.$$

Then we can rewrite the first equation and solve it:

$x^2+(1-x)^2=5,$

$x^2+1-2x+x^2=5,$  (we opened parentheses)

$2x^2-2x-4=0,$  (we added the similar terms)

$x^2-x-2=0,$  (we divided the equation by 2)

$x^2-2x+x-2=0,$

$x(x-2)+1(x-2)=0,$

(x-2)(x+1)=0,

$x-2=0$  or $x+1=0,$

$x_1=2$ ,  $x_2=-1.$

Now we can find the y:

$y_1=1-x_1=1-2=-1,$

$y_2=1-x_2=1-(-1)=2.$

So, we have two points:  (2,-1) and (-1,2).

The graphical representation.

The graphical representation of the first equation is circle with the center in (0;0) and radius , the graphical representation of the second equation is straight line $y=1-x$ .

The graphical representation of solution are points of intersection: (2,-1) and (-1,2).

Answer: (2,-1) and (-1,2).