x^2 + y^2 = 25
3x + y = 15
Find the two points of intersection.
I don't just want the answer, if you could tell me how to find it, it would save my life. The sqaure root is making it really hard for me to understand.
from the second,y=15-3x
then square it...
225-90x+9x^2 check that, and put it in for y^2 in the first equation, rearrange it in quadratic form, and solve for x using the quadratic formula.
To find the points of intersection, we need to solve the system of equations:
Equation 1: x^2 + y^2 = 25
Equation 2: 3x + y = 15
Let's solve Equation 2 for y:
y = 15 - 3x
Substitute this value of y into Equation 1:
x^2 + (15 - 3x)^2 = 25
Expand and simplify Equation 1:
x^2 + (225 - 90x + 9x^2) = 25
10x^2 - 90x + 200 = 0
Divide the entire equation by 10 for simplification:
x^2 - 9x + 20 = 0
Now, we need to solve this quadratic equation for x. You can factor this equation or use the quadratic formula.
Factoring method:
(x - 5)(x - 4) = 0
This gives us two possible values for x: x = 4 or x = 5.
Now that we have the x-values, we can substitute them back into Equation 2 to find the corresponding y-values.
For x = 4, y = 15 - 3(4) = 15 - 12 = 3. So, one point of intersection is (4, 3).
For x = 5, y = 15 - 3(5) = 15 - 15 = 0. So, the other point of intersection is (5, 0).
Therefore, the two points of intersection are (4, 3) and (5, 0).