Systems of equations & inequalities
The idea
A system gathers several conditions that must hold at once, and solving it means finding every point that satisfies all of them simultaneously. You have solved pairs of linear equations; high school widens the cast to linear–quadratic systems, where a line meets a parabola, and to systems of inequalities, whose solution is a shaded region rather than a point. Geometrically each equation draws a curve and each solution is an intersection — so a line can meet a parabola twice, once, or never.
Substitution is the universal tool: solve one equation for a variable, substitute into the other, and two equations in two unknowns collapse into one equation in one unknown. Elimination — adding multiples of equations to cancel a variable — is quicker when everything is linear. For inequalities, graph each boundary, pick the correct side with a test point, and intersect the shaded regions. The misconception to avoid is stopping once x is found: a system's solution is a complete point, so each x must be paired with its own y by substituting back.
Worked example
Find all points where the line y = 2x − 1 intersects the parabola y = x² − 4x + 7.
- Both formulas equal y at an intersection, so set them equal: x² − 4x + 7 = 2x − 1. Substitution has collapsed the system into a single equation in x.
- Move everything to one side: x² − 6x + 8 = 0. This factors as (x − 2)(x − 4) = 0, so x = 2 or x = 4.
- Each x needs its partner y; use the simpler equation, the line: x = 2 gives y = 2 × 2 − 1 = 3, and x = 4 gives y = 2 × 4 − 1 = 7. The candidate points are (2, 3) and (4, 7).
- Confirm both points in the parabola: 2² − 8 + 7 = 3 and 4² − 16 + 7 = 7, matching the line's values. Two distinct intersections means the line cuts clean through the parabola.
Answer. The line meets the parabola at the two points (2, 3) and (4, 7).
Check your understanding
- What does the discriminant of the combined equation reveal about the three ways a line can meet a parabola?
- Why does adding a multiple of one equation to another leave the solution set of a system unchanged?
- How does the solution of a system of inequalities differ in kind from the solution of a system of equations, and how would you report each one?
- What features of a system signal that substitution will be easier than elimination, or the other way around?
Build the foundations first
Systems of equations & inequalities builds on these concepts. If any feel shaky, start there.