SAT-38: Systems of Non-Linear Equations (Linear–Quadratic)

SAT-38: Systems of Non-Linear Equations (Linear–Quadratic)

Description: Sometimes the SAT gives a line and a parabola together and asks where they meet. The solutions are the intersection points. The reliable method is substitution.

The method (substitution)

1. Solve the linear equation for y (or x).
2. Substitute that into the quadratic equation.
3. You now have one quadratic in a single variable — set it to 0 and solve.
4. Plug each x back into the line to get its y.

(Oʻzbekcha: chiziqli tenglamani y uchun yeching, soʻngra uni kvadrat tenglamaga qoʻying.)

Worked Example

Solve the system: y = x + 1 and y = x² − 1.

• Substitute: x + 1 = x² − 1.
• Rearrange to 0: x² − x − 2 = 0 → (x − 2)(x + 1) = 0 → x = 2 or x = −1.
• Back into the line y = x + 1: x = 2 → y = 3; x = −1 → y = 0.
• Intersection points: (2, 3) and (−1, 0).

(Oʻzbekcha: ikkita yechim — chiziq parabolani ikki nuqtada kesib oʻtadi.)

How many intersections? (use the discriminant)

After substituting you get a quadratic. Its discriminant (b² − 4ac, from SAT-34) tells you how many times the graphs meet:

• discriminant > 0 → two intersection points,
• discriminant = 0 → one point (the line is tangent / just touches),
• discriminant < 0 → no intersection (they never meet).

Tip: "for what value of k is the line tangent to the parabola?" means set the discriminant equal to 0. (Oʻzbekcha: "urinma" deganda diskriminant = 0 qilinadi.)

Practice

How many times does y = 2x − 3 meet y = x²?

Key words — Kalit soʻzlar

• System of equations — tenglamalar sistemasi
• Non-linear — chiziqli boʻlmagan
• Substitution — oʻrniga qoʻyish
• Intersection point — kesishish nuqtasi
• Linear equation — chiziqli tenglama
• Quadratic equation — kvadrat tenglama
• Discriminant — diskriminant
• Tangent — urinma
• Rearrange — qayta tartiblash

Summary

• Solve the line for y, substitute into the quadratic, solve the resulting quadratic.
• Each x-solution gives an intersection point (find y from the line).
• The discriminant counts intersections: positive → 2, zero → 1 (tangent), negative → 0.