What is the geometric interpretation of the system of equations y=y^2-x+1 y=3-2x

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The two equations given are y = y^2 - x + 1 and y = 3- 2x

Substituting y = 3 - 2x in y = y^2 - x + 1

=> 3 - 2x = (3 - 2x)^2 - x + 1

=> 3 - 2x = 9 + 4x^2 - 12x - x + 1

=> 4x^2 - 11x + 7 = 0

=> 4x^2 - 7x - 4x + 7 = 0

=> x(4x - 7) - 1(4x - 7) = 0

=> (x - 1)(4x - 7) = 0

=> x = 1 and x = 7/4

=> y = 1 and y = -1/2

The graphs of the two equations intersect each other at the points (1,1) and (7/4 , -1/2)

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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The geometric interpretation of the system means to check if the given line and parabola are intercepting in 2 points, one point or any point at all.

We'll solve the system putting the 1st equation equal to the 2nd one.

x^2 - x + 1 = 3 - 2x

We'll move all terms to the left:

x^2 - x + 1 - 3 + 2x = 0

We'll combine like terms:

x^2 + x - 2 = 0

We'll apply quadratic formula:

x1 = [-1+sqrt(1 + 8)]/2

x1 = (-1+3)/2

x1 = 1

x2 = -2

The y va;ues are:

y1 = 3 - 2x

y1 =  3 - 2

y1 = 1

y2 = 3 + 4

y2 = 7

The solutions of the system are the pairs: (1 , 1) and (-2 , 7).

Since the system has 2 solutions, that means that there are 2 intercepting points of the line and parabola and they are:  (1 , 1) and (-2 , 7).

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