We want to find the intersection points of the graphs given by the following system of equations:{y=2x−1x2+y2=1One of the intersection points is (0,−1).Find the other intersection point. Your answer must be exact.
Q. We want to find the intersection points of the graphs given by the following system of equations:{y=2x−1x2+y2=1One of the intersection points is (0,−1).Find the other intersection point. Your answer must be exact.
Given Equations: We are given the system of equations:1. y=2x−12. x2+y2=1We already know one intersection point is (0,−1). To find the other intersection point, we can substitute the expression for y from the first equation into the second equation.
Substitute and Expand: Substitute y=2x−1 into x2+y2=1: x2+(2x−1)2=1 Expand the squared term: x2+(4x2−4x+1)=1
Combine Like Terms: Combine like terms:x2+4x2−4x+1=15x2−4x+1=1Subtract 1 from both sides:5x2−4x=0
Factor and Solve for x: Factor out x:x(5x−4)=0Set each factor equal to zero:x=0 or 5x−4=0We already know that x=0 gives us the intersection point (0,−1), so we need to solve for the other value of x:5x−4=0
Find y-coordinate: Add 4 to both sides:5x=4Divide by 5:x=54
Calculate y: Now that we have the x-coordinate of the other intersection point, we need to find the corresponding y-coordinate. We can do this by substituting x=54 into the first equation y=2x−1:y=2(54)−1
Final Intersection Point: Calculate the value of y:y=58−1Convert 1 to a fraction with a denominator of 5:y=58−55y=58−5y=53
Final Intersection Point: Calculate the value of y: y=58−1 Convert 1 to a fraction with a denominator of 5: y=58−55 y=58−5 y=53We have found the x-coordinate and y-coordinate of the other intersection point to be (54,53). This is the exact value of the other intersection point.
More problems from Scale drawings: scale factor word problems