Digital SATMath: Advanced MathHard
A robotics engineer is designing a robot arm's movement. The arm's position (x, y) at time t (in seconds) is described by the system of equations: y = x^2 - 4x + 5 and y = 2x - 3. At what x-coordinate(s) does the robot arm's path intersect?
- Ax = 1 and x = 4
- Bx = 1 and x = 8
- Cx = 2 and x = 4
- Dx = 4 and x = 8
Show answer & explanationAnswer & explanation
Correct answer: C. x = 2 and x = 4
To find the intersection points, set the two expressions for y equal to each other: x^2 - 4x + 5 = 2x - 3. Rearrange the equation to form a standard quadratic equation: x^2 - 4x - 2x + 5 + 3 = 0, which simplifies to x^2 - 6x + 8 = 0. Factor the quadratic: (x - 2)(x - 4) = 0. This gives the solutions x = 2 and x = 4.
Why the other options are wrong
- A. This is a plausible distractor, perhaps from incorrect factoring.
- B. This could be a result of incorrect factoring or arithmetic.
- D. This implies a mistake in the quadratic formula or factoring.
Solving Systems of Nonlinear Equations (Substitution)
To solve a system of nonlinear equations by substitution, express one variable from one equation in terms of the other, then substitute this expression into the other equation. This often reduces the system to a single, solvable equation.
- Common for one equation to be linear, the other quadratic.
- Set expressions for the same variable equal to each other.
- Solving the resulting equation yields the x-coordinates (or y-coordinates) of intersection.
Memory trick: Set Y's Equal, Solve for X!