Graph the line given by 2x+y=1 and the circle given by x²+y²=10.Find all solutions to the system of equations. Verify your result both algebraically and graphically.

Accepted Solution

Answer:(1.8, -2.6) and (-1, 3)Step-by-step explanation:[tex]2x+y=1[/tex][tex]x^2+y^2=10[/tex]From the first equation[tex]y=1-2x[/tex]Applying to the second equation[tex]x^2+(1-2x)^2=10\\\Rightarrow x^2+1+4x^2-4x=10\\\Rightarrow 5x^2-4x-8=0[/tex]Solving the equation we get[tex]x=\frac{-\left(-4\right)+\sqrt{\left(-4\right)^2-4\cdot \:5\left(-9\right)}}{2\cdot \:5}, \frac{-\left(-4\right)-\sqrt{\left(-4\right)^2-4\cdot \:5\left(-9\right)}}{2\cdot \:5}\\\Rightarrow x=1.8, -1[/tex]At x = 1.8Applying in first equation[tex]2\times 1.8+y=1\\\Rightarrow y=1-3.6\\\Rightarrow y=-2.6[/tex]At x = -1Applying in first equation[tex]2\times -1+y=1\\\Rightarrow y=1+2\\\Rightarrow y=3[/tex]∴ The circle and line intersect at points (1.8, -2.6) and (-1, 3)