By solving the equations as a system, find the points common to the line with equation x-y=6 and the circle with equation x²+y²=26.graph the line and the circle to show those points.

The solutions to the system are the points (5, -1) and (1, -5)

Step-by-step explanation:

Hi there!

We have the following system of equations:

x - y = 6

x² + y² = 26

The points common to the line and the circle are those (x,y) values that satisfy both equations. So let´s take the first equation and solve it for x:

x - y = 6

add y to both sides of the equation

x = 6 + y

Now, let´s replace the x in the second equation:

x² + y² = 26

(6 + y)² + y² = 26

(6 + y)(6 + y) + y² = 26

36 + 12y + y² + y² = 26

subtract 26 to both sides of the equation

10 + 12y + 2y² = 0

Solve the quadratic equation using the quadratic formula:

a = 2

b = 12

c = 10

[-b ± √(b² - 4ac)] / 2a

The solutions to the quadratic equation are y = -5 and y = -1

Let´s calculate the x value:

x = 6 + y

For y = -5

x = 6 - 5 = 1

For y = -1

x = 6 - 1 = 5

Then, the solutions to the system and the points at which the line and the circle intersect are (5, -1) and (1, -5). Please, see the attached figure to corroborate this graphically.

Have a nice day!