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Zorluk: OrtaSystems of Linear and Non-Linear Equations

A parabola is defined by the equation y=x2−4x+3y = x^2 - 4x + 3, and a line is defined by the equation y=−x+7y = -x + 7. The parabola and the line intersect at two points in the standard (x,y)(x, y) coordinate plane. What is the distance between these two points of intersection?

  1. A
    5
  2. B
    34\sqrt{34}
  3. 525\sqrt{2}Cevap
  4. D
    323\sqrt{2}
  5. E
    50

Cevap

525\sqrt{2}
The correct answer is the value representing the straight-line distance between the two intersection points. Setting the equations equal to each other gives x2−3x−4=0x^2 - 3x - 4 = 0, which factors to (x−4)(x+1)=0(x - 4)(x + 1) = 0. This yields x=4x = 4 and x=−1x = -1. Evaluating these in y=−x+7y = -x + 7 yields the points (4,3)(4, 3) and (−1,8)(-1, 8). The distance between them is (4−(−1))2+(3−8)2=25+25=50=52\sqrt{(4 - (-1))^2 + (3 - 8)^2} = \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2}.

Adım Adım Çözüm

1
Set the equations of the parabola and line equal to each other to find the xx-coordinates of the intersection points.
x2−4x+3=−x+7x^2 - 4x + 3 = -x + 7
Intersection points must satisfy both equations simultaneously.
2
Rearrange the equation into standard quadratic form and factor it to solve for xx.
x2−3x−4=0  ⟹  (x−4)(x+1)=0  ⟹  x=4x^2 - 3x - 4 = 0 \implies (x - 4)(x + 1) = 0 \implies x = 4 or x=−1x = -1
Factoring the quadratic equation gives the roots which correspond to the xx-coordinates of the intersection points.
3
Substitute the xx-values back into the linear equation to determine the corresponding yy-coordinates.
For x=4x = 4, y=−(4)+7=3y = -(4) + 7 = 3, yielding point (4,3)(4, 3). For x=−1x = -1, y=−(−1)+7=8y = -(-1) + 7 = 8, yielding point (−1,8)(-1, 8).
Substituting into the simpler linear equation provides the yy-coordinates of the intersection points.
4
Apply the distance formula to calculate the distance between (4,3)(4, 3) and (−1,8)(-1, 8).
d=(4−(−1))2+(3−8)2=52+(−5)2=25+25=50=52d = \sqrt{(4 - (-1))^2 + (3 - 8)^2} = \sqrt{5^2 + (-5)^2} = \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2}
The distance formula calculates the straight-line distance between two coordinates in the coordinate plane.

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Systems of Linear and Non-Linear Equations
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Systems of Linear and Non-Linear Equations

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