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Zorluk: KolaySystems of Linear and Non-Linear Equations

A parabola is defined by the equation y=2x2−5x+1y = 2x^2 - 5x + 1 and a line is defined by the equation y=x−3y = x - 3. If the parabola and the line intersect at the points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2), what is the value of y1+y2y_1 + y_2?

Cevap: -3

Cevap

The sum of the y-coordinates of the intersection points is -3.
By setting the two equations equal to each other, we obtain a quadratic equation in terms of xx. Solving this equation gives the x-coordinates of the intersection points. Substituting these x-values back into the linear equation yields the corresponding y-coordinates. Summing these y-coordinates gives the final value of −3-3.

Adım Adım Çözüm

1
Equate the equations for the line and the parabola to find the x-coordinates of their intersection points.
2x2−5x+1=x−32x^2 - 5x + 1 = x - 3
Since both equations are equal to yy, their right-hand sides must be equal at the points of intersection.
2
Rearrange the equation into standard quadratic form ax2+bx+c=0ax^2 + bx + c = 0.
2x2−6x+4=02x^2 - 6x + 4 = 0
Grouping all terms on one side of the equation allows us to solve for $x.
3
Divide the entire equation by 2 to simplify.
x2−3x+2=0x^2 - 3x + 2 = 0
Simplifying the quadratic equation makes factoring easier.
4
Factor the quadratic equation to solve for xx.
(x−1)(x−2)=0(x - 1)(x - 2) = 0, yielding x=1x = 1 and x=2x = 2.
Finding the roots of the quadratic equation gives the x-coordinates of the intersection points.
5
Substitute each x-coordinate back into the linear equation y=x−3y = x - 3 to find the corresponding y-coordinates.
For x1=1x_1 = 1: y1=1−3=−2y_1 = 1 - 3 = -2. For x2=2x_2 = 2: y2=2−3=−1y_2 = 2 - 3 = -1.
The intersection points must satisfy both equations in the system.
6
Calculate the sum of the two y-coordinates, y1+y2y_1 + y_2.
y1+y2=−2+(−1)=−3y_1 + y_2 = -2 + (-1) = -3
The question asks for the value of the sum of the y-coordinates.

Anahtar Kavram

Solving systems of linear and quadratic equations by substitution
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Konular

Systems of Linear and Non-Linear Equations

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