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We want to find the intersection points of the graphs given by the following system of equations:

{[y=2x-1],[x^(2)+y^(2)=1]:}
One of the intersection points is 
(0,-1).
Find the other intersection point. Your answer must be exact.

We want to find the intersection points of the graphs given by the following system of equations:\newline{y=2x1x2+y2=1 \left\{\begin{array}{l} y=2 x-1 \\ x^{2}+y^{2}=1 \end{array}\right. \newlineOne of the intersection points is (0,1) (0,-1) .\newlineFind the other intersection point. Your answer must be exact.

Full solution

Q. We want to find the intersection points of the graphs given by the following system of equations:\newline{y=2x1x2+y2=1 \left\{\begin{array}{l} y=2 x-1 \\ x^{2}+y^{2}=1 \end{array}\right. \newlineOne of the intersection points is (0,1) (0,-1) .\newlineFind the other intersection point. Your answer must be exact.
  1. Given Equations: We are given the system of equations:\newline11. y=2x1y = 2x - 1\newline22. x2+y2=1x^2 + y^2 = 1\newlineWe already know one intersection point is (0,1)(0, -1). To find the other intersection point, we can substitute the expression for yy from the first equation into the second equation.
  2. Substitute and Expand: Substitute y=2x1y = 2x - 1 into x2+y2=1x^2 + y^2 = 1:
    x2+(2x1)2=1x^2 + (2x - 1)^2 = 1
    Expand the squared term:
    x2+(4x24x+1)=1x^2 + (4x^2 - 4x + 1) = 1
  3. Combine Like Terms: Combine like terms:\newlinex2+4x24x+1=1x^2 + 4x^2 - 4x + 1 = 1\newline5x24x+1=15x^2 - 4x + 1 = 1\newlineSubtract 11 from both sides:\newline5x24x=05x^2 - 4x = 0
  4. Factor and Solve for xx: Factor out xx:x(5x4)=0x(5x - 4) = 0Set each factor equal to zero:x=0 or 5x4=0x = 0 \text{ or } 5x - 4 = 0We already know that x=0x = 0 gives us the intersection point (0,1)(0, -1), so we need to solve for the other value of xx:5x4=05x - 4 = 0
  5. Find y-coordinate: Add 44 to both sides:\newline5x=45x = 4\newlineDivide by 55:\newlinex=45x = \frac{4}{5}
  6. Calculate yy: Now that we have the xx-coordinate of the other intersection point, we need to find the corresponding yy-coordinate. We can do this by substituting x=45x = \frac{4}{5} into the first equation y=2x1y = 2x - 1:\newliney=2(45)1y = 2\left(\frac{4}{5}\right) - 1
  7. Final Intersection Point: Calculate the value of yy:y=851y = \frac{8}{5} - 1Convert 11 to a fraction with a denominator of 55:y=8555y = \frac{8}{5} - \frac{5}{5}y=855y = \frac{8 - 5}{5}y=35y = \frac{3}{5}
  8. Final Intersection Point: Calculate the value of yy:
    y=851y = \frac{8}{5} - 1
    Convert 11 to a fraction with a denominator of 55:
    y=8555y = \frac{8}{5} - \frac{5}{5}
    y=855y = \frac{8 - 5}{5}
    y=35y = \frac{3}{5}We have found the xx-coordinate and yy-coordinate of the other intersection point to be (45,35)(\frac{4}{5}, \frac{3}{5}). This is the exact value of the other intersection point.

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