Question

Graph the line given by 3x+4y=25, and the circle given by x²+y²=25.Find all solutions to the system of equations. Verify your result both algebraically and graphically.

Answer 1

(3,4)

Explanation:

`3x+4y=25`

`x^2+y^2=25`

From the first equation

`x=(25-4y)/(3)`

Applying to the second equation

`\left((25-4y)/(3)\right)^2+y^2=0\\\Rightarrow (625+16y^2-200y+9y^2)/(9)=25\\\Rightarrow 625+16y^2-200y+9y^2=9* 25\\\Rightarrow 25y^2-200y+625=225\\\Rightarrow 25y^2-200y+400=0\\\Rightarrow y^2-8y+16=0`

`y_(1,\:2)=(-\left(-8\right)\pm √(\left(-8\right)^2-4\cdot \:1\cdot \:16))/(2\cdot \:1)\\\Rightarrow y=4`

y=4.

Point on the circle

`x=√(25-16)=3`

So, the line will intersect at the point (3,4)