Where nonlinear models meet · SAT Math

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?

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

Answer: 3

Compare the other answer choices

Return only the larger intersection input.

Return the sum of the intersection inputs.

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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