SAT Math — systems with a nonlinear equation: substitution and counting intersections

The digital SAT's Advanced Math domain includes systems of equations in two variables in which at least one equation is nonlinear. The questions ask for a solution, for a value built from the solutions, for the number of solutions, or for the constant that produces a given number of solutions. This material covers all four types with a wider range of curves than a line meeting a parabola.

You solve a circle and a line by substitution, two parabolas by setting their expressions equal, a hyperbola y = 6/x and a line by clearing the fraction, and the pair xy = 12, x + y = 7 with the identity x² + y² = (x + y)² − 2xy. An exponential curve and a line are compared by testing values and reasoning about how many times a line can meet an upward-bending curve. A context question finds when a thrown ball and a rising drone are at the same height.

Counting questions use the discriminant of the combined equation: the constant that makes a line tangent to a parabola, the positive intercept of a line tangent to a circle, the values of a slope for which a line misses a parabola, and a system with a double root. A horizontal line and a downward parabola are compared through the vertex, and a circle and a parabola turn out to meet at three points, because one of the y-values gives only x = 0.

The material offers a quiz of twelve four-option questions, each with an explanation that shows the method and names the slip behind each wrong option, and a set of flashcards that collects the rules to remember.

The content is based on the College Board's published skill descriptions for the Advanced Math domain of the digital SAT Math section. It is independent practice and is not produced or endorsed by the College Board.

  • Solve systems of a linear and a nonlinear equation by substitution
  • Solve systems of two nonlinear equations, including circles and parabolas
  • Use the discriminant to count solutions and to find constants for tangency or no solution
  • Relate the solutions of a system to the intersection points of its graphs
  • Model and solve a real-world situation with a nonlinear system

Practice material written by Zestly, based on the College Board's published description of the digital SAT Math section (Advanced Math domain: systems of equations in two variables with a nonlinear equation), as given on satsuite.collegeboard.org and in the Assessment Framework for the Digital SAT Suite.

Sample question

Which ordered pair $(x, y)$ is a solution of the system of equations $x^2 + y^2 = 25$ and $y = x + 1$?

See the answer

$(3, 4)$

Substitute $y = x + 1$: $x^2 + (x + 1)^2 = 25$, so $2x^2 + 2x - 24 = 0$, $x^2 + x - 12 = 0$ and $(x + 4)(x - 3) = 0$. For $x = 3$, $y = 4$: the pair $(3, 4)$ satisfies both equations. The other solution is $(-4, -3)$. The pair $(4, 3)$ lies on the circle but not on the line, since $3 \neq 4 + 1$; $(0, 5)$ is also on the circle only; and $(-3, -2)$ is on the line but $9 + 4 = 13 \neq 25$.

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