what is the total number of points that graphs x^2+y^2= 16 and y=x have in common
To find the total number of points that the graphs of the equations x^2 + y^2 = 16 and y = x have in common, we need to solve the system of equations.
First, let's solve the equation x^2 + y^2 = 16 for y. We can rearrange the equation as y^2 = 16 - x^2 and take the square root of both sides to solve for y. Since it is a quadratic equation, we will have both positive and negative values for y.
Taking the square root, we have y = ±√(16 - x^2).
Next, we substitute y = x into this equation and solve for x.
x = ±√(16 - x^2)
We can square both sides of the equation to get rid of the square root:
x^2 = 16 - x^2
2x^2 = 16
Simplifying further, we have:
x^2 = 8
Taking the square root of both sides, we get:
x = ±√8
Simplifying the square root of 8, we have:
x = ±2√2
Now that we have the x-values, we can substitute them back into the equation y = x to find the corresponding y-values:
For x = 2√2, y = 2√2
For x = -2√2, y = -2√2
Therefore, the graph of x^2 + y^2 = 16 and y = x have two common points: (√8, √8) and (-√8, -√8).