Question 1

What is the elimination method for solving a system of equations?

Accepted Answer

Elimination works by adding or subtracting the two equations to cancel one variable. Example: x + y = 10 and x − y = 4. Adding gives 2x = 14, so x = 7, then y = 3. When coefficients do not match, multiply one or both equations by a constant first. For example, to eliminate x from 2x + 3y = 12 and 5x − y = 7, multiply the first by 5 and the second by 2, then subtract.

Question 2

What is the substitution method?

Accepted Answer

Substitution: solve one equation for one variable, then substitute that expression into the other equation. Example: y = 2x − 1 and 3x + y = 9. Substitute: 3x + (2x−1) = 9 → 5x = 10 → x = 2, y = 3. Substitution is most efficient when one equation is already solved for a variable. Elimination tends to be faster when coefficients are similar.

Question 3

When does a system have no solution?

Accepted Answer

A system has no solution when the two equations represent parallel lines — same slope, different y-intercepts. Algebraically, elimination will produce a contradiction like 0 = 5. Geometrically, parallel lines never intersect so there is no common (x, y). Example: x + y = 3 and x + y = 7 — subtracting gives 0 = 4, which is impossible.

Question 4

When does a system have infinitely many solutions?

Accepted Answer

Infinite solutions occur when both equations represent the same line — one is a multiple of the other. Elimination produces a trivially true statement like 0 = 0. Every point on the line satisfies both equations. The solution is expressed as a parametric set: for example, x = t, y = 3 − t for all real t. This is called a dependent system.

Question 5

How do I solve a 3×3 system of equations?

Accepted Answer

For three equations with three unknowns, use Gaussian elimination: create a triangular system by systematically eliminating variables. First eliminate x from equations 2 and 3, then eliminate y from equation 3. Back-substitute to find each variable. Alternatively, use matrix methods — express as Ax = b and compute x = A⁻¹b if the determinant of A is non-zero.

Question 6

What does the determinant of the coefficient matrix tell you?

Accepted Answer

For the system a₁x + b₁y = c₁, a₂x + b₂y = c₂, the coefficient matrix determinant is D = a₁b₂ − a₂b₁. If D ≠ 0, there is exactly one solution (x = (c₁b₂ − c₂b₁)/D, y = (a₁c₂ − a₂c₁)/D — Cramer's Rule). If D = 0, the lines are either parallel (no solution) or identical (infinite solutions) — check the constants to distinguish.

Type	Determinant D	Solutions	Geometry	Example
Consistent & independent	D ≠ 0	Exactly 1	Lines intersect at 1 point	x+y=5, x−y=1 → (3,2)
Inconsistent	D = 0	None	Lines are parallel	x+y=3, x+y=7
Dependent	D = 0	Infinite	Lines are identical	x+y=3, 2x+2y=6

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