Systems Involving Linear and Nonlinear Equations

7.7: Systems Involving Linear and Nonlinear Equations. Original editorial SAT preparation with three worked examples, ten flashcards, and thirty practice questions. Educator-review draft.

Lesson 7 of 83 parts Practice Flashcards

1/3

Line and parabola intersections

Objective: Substitute a line into a quadratic relation.

Prerequisite skills: substitution and quadratic equations. If y is given by two expressions in x, equate them. Solve the resulting quadratic, then use the line equation to obtain each y-coordinate. Two distinct x-roots give two points when y is uniquely determined. Common error: finding x-values but pairing them with unrelated y-values. Restrictions and interpretation: Check ordered pairs in both originals; a two-root equation can give two intersection points.

Formulas

$y=f(x), y=g(x) ⇒ f(x)=g(x)$

Worked examples

Try itFor y = x² and y = 23x, what is the greater x-coordinate of an intersection?Show answer

Set x²=23x, giving x(x−23)=0. The x-coordinates are 0 and 23; the greater is 23, with y=529. Therefore the requested answer is 23.

Swipe sideways to move between sections

Line and parabola intersections

Objective: Substitute a line into a quadratic relation.

Formulas

$y=f(x), y=g(x) ⇒ f(x)=g(x)$

Worked examples

Try itFor y = x² and y = 23x, what is the greater x-coordinate of an intersection?Show answer

Set x²=23x, giving x(x−23)=0. The x-coordinates are 0 and 23; the greater is 23, with y=529. Therefore the requested answer is 23.

A line through a shifted parabola

Objective: Solve a nonlinear system after substitution.

A horizontal line y=k intersects a parabola by setting the quadratic output equal to k. Use both square-root branches when the resulting square equals a positive value. Common error: counting only the right-side intersection. Restrictions and interpretation: Check ordered pairs in both originals; a two-root equation can give two intersection points.

Formulas

$(x−h)² = r² ⇒ x=h±r$

Worked examples

Try itThe system y = (x − 25)² + 5 and y = 534 has two intersections. What is the sum of their x-coordinates?Show answer

Set (x−25)²+5=534; then (x−25)²=529. Thus x=25±23, and the sum is 2(25)=50. Therefore the requested answer is 50.

Systems with a squared relation

Objective: Use substitution while preserving multiple sign possibilities.

When one equation expresses x linearly in y and another contains y², substitute to obtain a quadratic in y. Recover the companion coordinate from the same branch and check both equations. Common error: choosing a root but forgetting to calculate its companion coordinate. Restrictions and interpretation: Check ordered pairs in both originals; a two-root equation can give two intersection points. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.

Formulas

$x=y+c and x=y² ⇒ y²−y−c=0$

Worked examples

Try itFor x = y + 600 and x = y², what is the larger y-coordinate of a solution?Show answer

Substitute x=y²: y²=y+600, or y²−y−600=0. Factor (y−25)(y+24)=0. The larger y is 25; then x=625, which satisfies both equations. Therefore the requested answer is 25.

You finished the reading!

Lock it in with a few practice questions, then move on to the next lesson.

Practice Next lesson

Previous lessonFunction Transformations and Equivalent Representations Next lessonNonlinear Parameters and Number of Intersections