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Difficulty: MediumSystems of Linear and Non-Linear Equations

The elevation yy (in meters) of a roller coaster track is modeled by the equation y=(x−3)2−4y = (x - 3)^2 - 4, where xx represents the horizontal distance (in meters) from the start of the ride. A straight support beam is designed such that the height of the track is 7 meters less than 7 times the horizontal distance. The support beam connects to the roller coaster track at two points, (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2). What is the value of x1y1+x2y2x_1 y_1 + x_2 y_2?

  1. A
    126
  2. B
    -42
  3. 924Answer
  4. D
    336
  5. E
    2664

Answer

The value of the expression is 924, which corresponds to the coordinates of the two connection points (1,0)(1, 0) and (12,77)(12, 77).
By translating the verbal relationship to the equation y=7x−7y = 7x - 7, we can equate it to the track's equation: (x−3)2−4=7x−7(x - 3)^2 - 4 = 7x - 7. Expanding and simplifying yields x2−13x+12=0x^2 - 13x + 12 = 0, which factors as (x−1)(x−12)=0(x - 1)(x - 12) = 0. This gives the horizontal intersection coordinates x1=1x_1 = 1 and x2=12x_2 = 12. Substituting these coordinates back into the linear equation gives the vertical coordinates y1=0y_1 = 0 and y2=77y_2 = 77. Evaluating the final expression yields (1)(0)+(12)(77)=924(1)(0) + (12)(77) = 924.

Step-by-Step Solution

1
Translate the verbal description of the support beam's height into an algebraic equation.
y=7x−7y = 7x - 7
The description states that the height (yy) is '7 meters less than 7 times the horizontal distance (xx)', which translates to 7x−77x - 7.
2
Set the two equations for yy equal to each other to find the xx-coordinates of the intersection points.
(x−3)2−4=7x−7⇒x2−6x+5=7x−7⇒x2−13x+12=0(x - 3)^2 - 4 = 7x - 7 \Rightarrow x^2 - 6x + 5 = 7x - 7 \Rightarrow x^2 - 13x + 12 = 0
At the points of intersection, the elevation of the track and the support beam must be equal.
3
Solve the quadratic equation for xx by factoring.
(x−1)(x−12)=0⇒x1=1(x - 1)(x - 12) = 0 \Rightarrow x_1 = 1 and x2=12x_2 = 12
Factoring the quadratic expression allows us to identify its roots directly.
4
Substitute the xx-values back into the linear equation to find the corresponding yy-coordinates.
For x1=1x_1 = 1, y1=7(1)−7=0y_1 = 7(1) - 7 = 0. For x2=12x_2 = 12, y2=7(12)−7=77y_2 = 7(12) - 7 = 77. The intersection points are (1,0)(1, 0) and (12,77)(12, 77).
The intersection points must satisfy both equations in the system.
5
Calculate the required expression x1y1+x2y2x_1 y_1 + x_2 y_2.
(1)(0)+(12)(77)=0+924=924(1)(0) + (12)(77) = 0 + 924 = 924
This evaluates the target expression using the coordinates found in the previous steps.

Key Concept

Solving systems of linear and non-linear (quadratic) equations using substitution and factoring.
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Topics

Systems of Linear and Non-Linear Equations

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