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Difficulty: MediumNonlinear Systems of Equations
y=x2−x−72x−y=−3\begin{aligned} y &= x^2 - x - 7 \\ 2x - y &= -3 \end{aligned}

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

  1. A
    18
  2. -3Answer
  3. C
    -2
  4. D
    0

Answer

The correct answer is -3.
The correct answer is −3-3. To find this, substitute the expression for yy from the first equation into the second equation, which yields 2x−(x2−x−7)=−32x - (x^2 - x - 7) = -3. Distributing the negative sign and combining like terms gives −x2+3x+7=−3-x^2 + 3x + 7 = -3. Setting the equation to zero results in x2−3x−10=0x^2 - 3x - 10 = 0. Factoring the quadratic expression gives (x−5)(x+2)=0(x - 5)(x + 2) = 0, which means x=5x = 5 or x=−2x = -2. Since the question specifies that x<0x < 0, we select x=−2x = -2. Substituting x=−2x = -2 back into the linear equation gives y=2(−2)+3=−1y = 2(-2) + 3 = -1. Finally, calculating x+yx + y gives −2+(−1)=−3-2 + (-1) = -3.

Step-by-Step Solution

1
Substitute the expression for yy from the first equation into the second equation.
2x−(x2−x−7)=−32x - (x^2 - x - 7) = -3
This allows us to eliminate yy and solve for xx in a single variable quadratic equation.
2
Distribute the negative sign and simplify the equation.
−x2+3x+7=−3-x^2 + 3x + 7 = -3
Expanding the parentheses correctly preserves the signs of the quadratic terms.
3
Rearrange the terms to set the quadratic equation to zero.
x2−3x−10=0x^2 - 3x - 10 = 0
Writing the equation in standard form (ax2+bx+c=0ax^2 + bx + c = 0) allows us to factor it.
4
Factor the quadratic expression.
(x−5)(x+2)=0(x - 5)(x + 2) = 0
Finding factors that multiply to −10-10 and add to −3-3 gives us the solutions for xx.
5
Solve for xx and apply the constraint x<0x < 0.
x=−2x = -2
The equation yields x=5x = 5 or x=−2x = -2. The constraint x<0x < 0 means we must choose x=−2x = -2.
6
Substitute x=−2x = -2 back into one of the original equations to solve for yy.
y=2(−2)+3=−1y = 2(-2) + 3 = -1
We need the corresponding yy-value to find the value of x+yx + y.
7
Calculate the value of x+yx + y.
x+y=−2+(−1)=−3x + y = -2 + (-1) = -3
This provides the final required value specified in the question.

Key Concept

Solving a system of one linear equation and one quadratic equation using substitution.
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Topics

Nonlinear Systems of Equations

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