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Zorluk: KolaySystems of Linear and Non-Linear Equations

A line and a parabola intersect at a point (x,y)(x, y) in the first quadrant of the standard (x,y)(x, y) coordinate plane. If the equation of the line is y=x+1y = x + 1 and the equation of the parabola is y=(x−1)2y = (x - 1)^2, what is the value of x+yx + y?

  1. A
    3
  2. B
    5
  3. C
    1
  4. 7Cevap
  5. E
    9

Cevap

7
To find the intersection point, we set the two equations equal to each other: (x−1)2=x+1(x - 1)^2 = x + 1. Expanding the left side gives x2−2x+1=x+1x^2 - 2x + 1 = x + 1. Subtracting x+1x + 1 from both sides yields x2−3x=0x^2 - 3x = 0. Factoring gives x(x−3)=0x(x - 3) = 0, so x=0x = 0 or x=3x = 3. The first quadrant requires positive coordinates, so we choose x=3x = 3. Substituting this back into either equation gives y=3+1=4y = 3 + 1 = 4. Thus, the intersection point is (3,4)(3, 4), and the sum of the coordinates is 3+4=73 + 4 = 7.

Adım Adım Çözüm

1
Set the two equations equal to find the x-coordinates of the intersection points.
(x−1)2=x+1(x - 1)^2 = x + 1
Since both equations are equal to yy, their right-hand sides must be equal at the points of intersection.
2
Expand the quadratic expression and simplify the equation.
x2−2x+1=x+1  ⟹  x2−3x=0x^2 - 2x + 1 = x + 1 \implies x^2 - 3x = 0
Expanding the binomial (x−1)2(x-1)^2 yields x2−2x+1x^2 - 2x + 1. Subtracting xx and 11 from both sides simplifies the equation to standard quadratic form.
3
Factor the quadratic equation to solve for xx.
x(x−3)=0  ⟹  x=0x(x - 3) = 0 \implies x = 0 or x=3x = 3
Factoring out the greatest common factor xx allows us to find the roots of the equation.
4
Determine which solution lies in the first quadrant and calculate the corresponding y-coordinate.
x=3  ⟹  y=3+1=4x = 3 \implies y = 3 + 1 = 4
The first quadrant requires both coordinates to be strictly positive. The solution x=0x = 0 gives (0,1)(0, 1), which is on the y-axis. Thus, we must use x=3x = 3, which gives (3,4)(3, 4).
5
Calculate the sum x+yx + y.
3+4=73 + 4 = 7
The question asks for the value of x+yx + y for the first-quadrant intersection point.

Anahtar Kavram

Solving systems of linear and non-linear equations by substitution, factoring quadratic equations, and applying coordinate plane quadrant constraints.
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Systems of Linear and Non-Linear Equations

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