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Advanced Math / LESSON 16

Where nonlinear models meet · SAT Math

12 min + practice · Original foundation lesson

A nonlinear system still requires simultaneous agreement. When both equations express y, equate their right sides to find candidate inputs. Then recover the outputs and apply domain or sign conditions. A graph can suggest intersection locations, but algebra gives exact values and distinguishes one coordinate from the other.

Inputs and outputs are different targets

For y=x² and y=7x−10, equating outputs gives x²−7x+10=0=(x−2)(x−5). The input gap is 5−2=3. The outputs are four and 25, whose sum is 29. Reporting seven would add the inputs, answering a third question. Check each pair in both equations: (2,4) and (5,25) satisfy both.

Use structure for a sum-product system

If positive x,y obey x+y=10 and xy=21, then (x−y)²=100−84=16. With x>y, the difference is four. The numbers are seven and three. Taking both square-root signs without reading the ordering condition would leave the requested difference ambiguous.

A circle meeting a line

For x²+y²=25 and y=3, substitution gives x²=16. There are two points, (4,3) and (−4,3). A request for the positive input selects four, but it does not make the other intersection cease to exist. If y were six, the resulting x² would be negative and no real intersections would occur.

Transfer and check

Replace the line by y=5. The circle now has one intersection at (0,5), where the line is tangent. Explain why x²=0 has one distinct root, whereas x²=16 has two. The number of intersections depends on the constraints, not just the degree of one equation.

WORKED EXAMPLE

The graphs y=x² and y=9x−18 intersect at two points. What is the difference between the larger and smaller x-coordinates?

  1. Equate outputs: x²−9x+18=0.
  2. Factor as (x−3)(x−6)=0, giving inputs 3 and 6.
  3. The requested difference is 6−3=3; both points also satisfy the line.
Answer: 3
Compare the other answer choices
6

Return only the larger intersection input.

9

Return the sum of the intersection inputs.

18

Return the product of the intersection inputs.

Try it, then explain it.

Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.

Practice this skill

Sources & authorship

Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.

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