Examkin

Question

Difficulty: MediumNonlinear Systems of Equations
y=3x2−4x−5y−2x=4\begin{aligned} y &= 3x^2 - 4x - 5 \\ y - 2x &= 4 \end{aligned}

If (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) are the solutions to the system of equations above, and y1>y2y_1 > y_2, what is the value of x1+y2x_1 + y_2?

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

Answer

5
To find the solutions to the system of equations, substitute the expression for yy from the second equation into the first equation. First, rewrite the second equation as y=2x+4y = 2x + 4. Substituting this into the first equation yields 2x+4=3x2−4x−52x + 4 = 3x^2 - 4x - 5. Rearranging terms to set the equation to zero gives 3x2−6x−9=03x^2 - 6x - 9 = 0. Dividing the entire equation by 3 simplifies it to x2−2x−3=0x^2 - 2x - 3 = 0. Factoring this quadratic equation gives (x−3)(x+1)=0(x - 3)(x + 1) = 0, which yields the solutions x1=3x_1 = 3 and x2=−1x_2 = -1. Next, find the corresponding yy-coordinates by substituting these xx-values back into the linear equation y=2x+4y = 2x + 4. For x=3x = 3, y1=2(3)+4=10y_1 = 2(3) + 4 = 10. For x=−1x = -1, y2=2(−1)+4=2y_2 = 2(-1) + 4 = 2. We are given that y1>y2y_1 > y_2, which confirms that (x1,y1)=(3,10)(x_1, y_1) = (3, 10) and (x2,y2)=(−1,2)(x_2, y_2) = (-1, 2). Finally, calculate x1+y2x_1 + y_2, which is 3+2=53 + 2 = 5.

Step-by-Step Solution

1
Rewrite the linear equation to express yy in terms of xx.
y=2x+4y = 2x + 4
This allows for substitution into the quadratic equation.
2
Substitute y=2x+4y = 2x + 4 into the quadratic equation and set to zero.
3x2−6x−9=03x^2 - 6x - 9 = 0
To find the xx-coordinates of the intersection points.
3
Simplify and factor the quadratic equation.
(x−3)(x+1)=0(x - 3)(x + 1) = 0, giving x=3x = 3 and x=−1x = -1.
To solve for the xx-values of the intersection points.
4
Substitute the xx-values back into the linear equation to find the corresponding yy-values.
For x=3x = 3, y=10y = 10. For x=−1x = -1, y=2y = 2. The intersection points are (3,10)(3, 10) and (−1,2)(-1, 2).
To find the complete coordinates of the intersection points.
5
Identify (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) using the condition y1>y2y_1 > y_2, and compute x1+y2x_1 + y_2.
Since 10>210 > 2, y1=10y_1 = 10 (with x1=3x_1 = 3) and y2=2y_2 = 2 (with x2=−1x_2 = -1). Then, x1+y2=3+2=5x_1 + y_2 = 3 + 2 = 5.
To compute the required target expression.

Key Concept

Solving a nonlinear system of equations by substituting a linear expression into a quadratic equation and solving the resulting quadratic equation.
Rate this question

Topics

Nonlinear Systems of Equations

More questions from this topic