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Zorluk: OrtaSystems of Linear and Non-Linear Equations

In the standard (x,y)(x, y) coordinate system, a line given by the equation y=2x+4y = 2x + 4 intersects a parabola given by the equation y=x2−2x−1y = x^2 - 2x - 1 at exactly two points. What is the sum of the yy-coordinates of these two points of intersection?

Cevap: 16

Cevap

16
The system of equations is solved by setting the equations equal to each other, yielding the quadratic equation x2−4x−5=0x^2 - 4x - 5 = 0. Solving for xx gives x=5x = 5 and x=−1x = -1. Substituting these values into the linear equation gives the y-coordinates 1414 and 22. The sum of these y-coordinates is 14+2=1614 + 2 = 16.

Adım Adım Çözüm

1
Equate the linear and quadratic equations to find the x-values of the intersection points.
x2−2x−1=2x+4x^2 - 2x - 1 = 2x + 4
At the points of intersection, the y-values of both equations must be equal.
2
Rearrange the equation to standard quadratic form.
x2−4x−5=0x^2 - 4x - 5 = 0
Grouping all terms on one side allows the quadratic equation to be solved.
3
Factor the quadratic equation to find the x-coordinates.
(x−5)(x+1)=0(x - 5)(x + 1) = 0, so x=5x = 5 or x=−1x = -1
The numbers that multiply to −5-5 and add up to −4-4 are −5-5 and 11.
4
Substitute the x-coordinates into the linear equation to find the y-coordinates.
For x=5x = 5: y=2(5)+4=14y = 2(5) + 4 = 14. For x=−1x = -1: y=2(−1)+4=2y = 2(-1) + 4 = 2.
Evaluating the linear equation is simpler than evaluating the quadratic equation.
5
Find the sum of the y-coordinates.
14+2=1614 + 2 = 16
The question asks for the sum of the y-coordinates of the two points of intersection.

Anahtar Kavram

Solving a system of linear and quadratic equations by substitution.

Alternatif Yöntem

Instead of solving for the individual intersection points, Vieta's formulas can be applied. The x-coordinates satisfy x2−4x−5=0x^2 - 4x - 5 = 0, so their sum is x1+x2=4x_1 + x_2 = 4. Since the points lie on the line y=2x+4y = 2x + 4, the sum of the y-coordinates is y1+y2=(2x1+4)+(2x2+4)=2(x1+x2)+8=2(4)+8=16y_1 + y_2 = (2x_1 + 4) + (2x_2 + 4) = 2(x_1 + x_2) + 8 = 2(4) + 8 = 16.
Tahmini Süre:1m 30s
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Konular

Systems of Linear and Non-Linear Equations

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