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Difficulty: EasySystems of Linear and Non-Linear Equations

In the standard (x,y)(x, y) coordinate plane, a circle is defined by the equation x2+y2=17x^2 + y^2 = 17 and a line is defined by the equation y=x−3y = x - 3. If (x,y)(x, y) represents the intersection point of the circle and the line that lies in the first quadrant, what is the value of x+yx + y?

  1. A
    1
  2. B
    3
  3. 5Answer
  4. D
    7
  5. E
    -5

Answer

The sum of the coordinates of the first-quadrant intersection point is 5.
The correct answer is the sum of the coordinates of the first-quadrant intersection point. By substituting y=x−3y = x - 3 into the circle's equation, we get x2+(x−3)2=17x^2 + (x - 3)^2 = 17. Expanding the squared term gives x2+x2−6x+9=17x^2 + x^2 - 6x + 9 = 17, which simplifies to 2x2−6x−8=02x^2 - 6x - 8 = 0. Dividing the entire equation by 2 yields x2−3x−4=0x^2 - 3x - 4 = 0. Factoring this quadratic equation gives (x−4)(x+1)=0(x - 4)(x + 1) = 0, which has solutions x=4x = 4 and x=−1x = -1. Because the intersection point must lie in the first quadrant, the xx-coordinate must be positive, so we choose x=4x = 4. Substituting x=4x = 4 back into the linear equation gives y=4−3=1y = 4 - 3 = 1. The sum of these coordinates is 4+1=54 + 1 = 5.

Step-by-Step Solution

1
Substitute the expression for yy from the linear equation into the circle's equation.
x2+(x−3)2=17x^2 + (x - 3)^2 = 17
This eliminates yy so that we can solve for xx.
2
Expand the squared binomial and combine like terms to write the equation in standard quadratic form.
2x2−6x−8=02x^2 - 6x - 8 = 0
Expanding (x−3)2(x - 3)^2 gives x2−6x+9x^2 - 6x + 9, and adding x2x^2 and subtracting 17 from both sides gives the quadratic equation.
3
Simplify the quadratic equation by dividing all terms by 2, then factor the resulting quadratic expression.
(x−4)(x+1)=0(x - 4)(x + 1) = 0
Dividing by 2 gives x2−3x−4=0x^2 - 3x - 4 = 0, which factors into (x−4)(x+1)=0(x - 4)(x + 1) = 0 since (−4)×1=−4(-4) \times 1 = -4 and −4+1=−3-4 + 1 = -3.
4
Solve for xx and select the positive solution since the point lies in the first quadrant.
x=4x = 4
The solutions are x=4x = 4 and x=−1x = -1. In the first quadrant, both coordinates must be positive, so we choose x=4x = 4.
5
Substitute x=4x = 4 back into the linear equation to find the corresponding yy-coordinate.
y=1y = 1
Using y=x−3y = x - 3 with x=4x = 4 gives y=4−3=1y = 4 - 3 = 1.
6
Calculate the sum of the coordinates x+yx + y.
4+1=54 + 1 = 5
The question asks for the value of x+yx + y.

Key Concept

Solving a system consisting of a linear equation and a quadratic circle equation by substitution, factoring the resulting quadratic equation, and applying quadrant constraints.
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Topics

Systems of Linear and Non-Linear Equations

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