Simultaneous Equations – Elimination, Substitution, and Geometric Interpretation

Seven flashcards summarizing elimination and substitution methods for simultaneous equations, along with their geometric interpretation.

What does the elimination method involve in solving simultaneous equations?

Aligning two equations and adding or subtracting them to eliminate one variable, making it easier to solve for the other variable.

What are the sequential steps for solving simultaneous equations by elimination?

1) Match coefficients by multiplying the equations if necessary. 2) Add or subtract the equations to eliminate one variable. 3) Solve the resulting single-variable equation. 4) Substitute back to find the other variable. 5) Check the solution in both original equations.

When is the substitution method generally required in solving simultaneous equations?

When one of the equations is quadratic (contains x² or y²), making elimination less straightforward.

How is the substitution method carried out for simultaneous equations?

Rearrange the linear equation to make x or y the subject, substitute this expression into the quadratic equation, solve for the remaining variable, then back-substitute to find the other variable.

How can you interpret simultaneous equations geometrically?

By sketching the graphs of both equations and counting their intersections, which reveal the number of solutions.

What do different numbers of intersections between two graphs indicate about simultaneous equations?

Two intersections → two solutions; one intersection (tangency) → one solution; no intersection → no solution.

What does it mean if a straight line is tangent to a curve in the context of simultaneous equations?

The line touches the curve at exactly one point, indicating a single solution to the system.