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Difficulty: MediumNonlinear Systems of Equations

In the system of equations below, kk is a constant.

y=3x2−5x+4y = 3x^2 - 5x + 4
y=7x+ky = 7x + k

If the system has exactly one real solution (x,y)(x, y), what is the value of kk?

  1. A
    88
  2. −8-8Answer
  3. C
    1616
  4. D
    4040

Answer

−8-8
To find the number of solutions for the system of equations, we set the equations equal to each other: 3x2−5x+4=7x+k3x^2 - 5x + 4 = 7x + k. Subtracting 7x7x and kk from both sides gives the standard quadratic equation 3x2−12x+(4−k)=03x^2 - 12x + (4 - k) = 0. For a quadratic equation to have exactly one real solution, its discriminant, b2−4acb^2 - 4ac, must equal 00. Substituting a=3a = 3, b=−12b = -12, and c=4−kc = 4 - k into the discriminant formula gives (−12)2−4(3)(4−k)=0(-12)^2 - 4(3)(4 - k) = 0. Simplifying this equation yields 144−12(4−k)=0144 - 12(4 - k) = 0, which simplifies to 144−48+12k=0144 - 48 + 12k = 0, or 96+12k=096 + 12k = 0. Solving for kk gives k=−8k = -8.

Step-by-Step Solution

1
Equate the two equations to eliminate yy and form a single quadratic equation.
3x2−5x+4=7x+k⇒3x2−12x+(4−k)=03x^2 - 5x + 4 = 7x + k \Rightarrow 3x^2 - 12x + (4 - k) = 0
To find the points of intersection of the system, we set the expressions for yy equal to each other.
2
Identify the coefficients of the quadratic equation in standard form ax2+bx+c=0ax^2 + bx + c = 0.
a=3a = 3, b=−12b = -12, and c=4−kc = 4 - k
These coefficients are needed to calculate the discriminant of the quadratic equation.
3
Set the discriminant Δ=b2−4ac\Delta = b^2 - 4ac equal to zero and solve for kk.
(−12)2−4(3)(4−k)=0⇒144−12(4−k)=0⇒12(4−k)=144⇒4−k=12⇒k=−8(-12)^2 - 4(3)(4 - k) = 0 \Rightarrow 144 - 12(4 - k) = 0 \Rightarrow 12(4 - k) = 144 \Rightarrow 4 - k = 12 \Rightarrow k = -8
A quadratic equation has exactly one real solution if and only if its discriminant is equal to zero.

Key Concept

Solving nonlinear systems of equations by setting them equal to each other and using the discriminant of the resulting quadratic equation to determine the number of solutions.
Estimated Time:1m 30s
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Topics

Nonlinear Systems of Equations

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