Solve systems by graphing linear and quadratic equations. Grade 11 Math 20-1.
Lesson 2.2 of Linear Functions & Systems of Equations in Math 20-1 — Alberta curriculum lessons.
What is a Linear-Quadratic System?. A linear-quadratic system is a pair of equations where one is a line and the other is a parabola. When you graph both on the same grid, the points where they intersect are the solutions.
There are three possible scenarios depending on how the line meets the parabola:
What is a Tangent Line?. A tangent line touches a parabola at exactly one point without crossing it. This corresponds to the one-solution case for a linear-quadratic system.
What is a Quadratic-Quadratic System?. A quadratic-quadratic system is a pair of equations where both are parabolas. There are four possible scenarios:
Infinitely Many Solutions. Infinite solutions occur when both equations describe the same parabola — they are identical, just written differently. Every point on the parabola satisfies both equations.
Follow these steps whenever you solve a system of equations using a graphing calculator:
Key Reminder — Always Verify. After finding an intersection point, always substitute the coordinates back into both original equations to confirm it satisfies them. A single verification error can cost you the question.
Setup: y + 3x^2 - 2x - 4 = 0 and y + x + 2 = 0. Step 1: Isolate y in both equations.
From equation 1:
y = -3x^2 + 2x + 4
From equation 2:
y = -x - 2
Step 2 and 3 — Find and Verify Intersections. Step 2: Graph both on the calculator. The two graphs intersect at (-1, -1) and (2, -4).
Step 3: Verify (-1, -1).
Equation 1: -1 + 3(-1)^2 - 2(-1) - 4 = -1 + 3 + 2 - 4 = 0 ✓
Equation 2: -1 + (-1) + 2 = 0 ✓
Verify (2, -4).
Equation 1: -4 + 3(2)^2 - 2(2) - 4 = -4 + 12 - 4 - 4 = 0 ✓
Equation 2: -4 + 2 + 2 = 0 ✓
Solutions: (-1, -1) and (2, -4)
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