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Difficulty: MediumNonlinear Systems of Equations

If (x,y)(x, y) is the solution to the system of equations below, what is the value of xx?

x−y=4x - y = 4
x2−y2=40x^2 - y^2 = 40

Answer: 7

Answer

The value of xx is 7.
The equation x2−y2=40x^2 - y^2 = 40 can be factored as (x−y)(x+y)=40(x-y)(x+y) = 40. Substituting x−y=4x-y = 4 yields 4(x+y)=404(x+y) = 40, which simplifies to x+y=10x+y = 10. Adding the two linear equations x−y=4x-y = 4 and x+y=10x+y = 10 eliminates yy, resulting in 2x=142x = 14, or x=7x = 7.

Step-by-Step Solution

1
Factor the second equation using the difference of squares identity.
(x−y)(x+y)=40(x-y)(x+y) = 40
To rewrite the quadratic expression in a form where the linear equation can be substituted.
2
Substitute the first equation x−y=4x-y = 4 into the factored expression.
4(x+y)=404(x+y) = 40, which simplifies to x+y=10x+y = 10
To find a simpler linear relation for the sum of the variables.
3
Add the equations x−y=4x-y = 4 and x+y=10x+y = 10.
2x=142x = 14
To eliminate the variable yy and solve for xx directly.
4
Solve for xx by dividing by 2.
x=7x = 7
To isolate the variable and find the final value of xx.

Key Concept

Solving a nonlinear system of equations by factoring a difference of squares and substituting.

Alternative Method

Express xx from the first equation as x=y+4x = y + 4 and substitute it into the second equation: (y+4)2−y2=40(y+4)^2 - y^2 = 40. Expanding and simplifying yields y2+8y+16−y2=40y^2 + 8y + 16 - y^2 = 40, which simplifies to 8y+16=40  ⟹  8y=24  ⟹  y=38y + 16 = 40 \implies 8y = 24 \implies y = 3. Substituting y=3y = 3 back into x=y+4x = y + 4 gives x=7x = 7.
Estimated Time:1m 15s
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Topics

Nonlinear Systems of Equations

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