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Difficulty: MediumNonlinear Systems of Equations

If (x,y)(x, y) is a solution to the system of equations below and x>0x > 0, what is the value of x+yx + y?

x2+y2=50x−y=6\begin{aligned} x^2 + y^2 &= 50 \\ x - y &= 6 \end{aligned}

Answer: 8

Answer

8
Solving the linear equation for xx gives x=y+6x = y + 6. Substituting this into the circle equation yields (y+6)2+y2=50(y + 6)^2 + y^2 = 50, which simplifies to 2y2+12y−14=02y^2 + 12y - 14 = 0. Dividing by 22 gives y2+6y−7=0y^2 + 6y - 7 = 0, which factors as (y+7)(y−1)=0(y + 7)(y - 1) = 0. This yields y=1y = 1 (with x=7x = 7) or y=−7y = -7 (with x=−1x = -1). The constraint x>0x > 0 requires selecting the solution (7,1)(7, 1). The sum of xx and yy for this solution is 7+1=87 + 1 = 8.

Step-by-Step Solution

1
Express xx in terms of yy using the linear equation.
x=y+6x = y + 6
This allows for substitution into the second equation to reduce the system to a single variable.
2
Substitute the expression for xx into the equation of the circle and expand.
2y2+12y−14=02y^2 + 12y - 14 = 0
Substituting x=y+6x = y + 6 into x2+y2=50x^2 + y^2 = 50 yields (y+6)2+y2=50(y + 6)^2 + y^2 = 50. Expanding gives y2+12y+36+y2=50y^2 + 12y + 36 + y^2 = 50.
3
Simplify and solve the quadratic equation by factoring.
y=1y = 1 or y=−7y = -7
Dividing the equation by 2 gives y2+6y−7=0y^2 + 6y - 7 = 0, which factors as (y+7)(y−1)=0(y + 7)(y - 1) = 0.
4
Calculate the corresponding values of xx for each yy.
For y=1y = 1, x=7x = 7; for y=−7y = -7, x=−1x = -1
Using the relation x=y+6x = y + 6 yields the coordinates of the two intersection points.
5
Apply the constraint x>0x > 0 and calculate x+yx + y.
x+y=8x + y = 8
Since x>0x > 0, the only valid solution is (7,1)(7, 1). The sum of the coordinates is 7+1=87 + 1 = 8.

Key Concept

Solving a nonlinear system of equations representing a circle and a line using substitution and factoring.
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Topics

Nonlinear Systems of Equations

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