Question 1

How do you solve a system of linear equations by substitution?

Accepted Answer

To solve by substitution: 1) Choose an equation and solve for one variable (preferably one with a coefficient of 1 or -1) in terms of the other variable. 2) Substitute this expression into the other equation to get a single-variable linear equation. 3) Solve for that variable. 4) Back-substitute the numerical value into your expression to find the remaining variable.

Question 2

When should I use elimination instead of substitution?

Accepted Answer

Use the elimination method when variable coefficients are already identical, opposite, or easily matched by multiplying equations by their Least Common Multiple (LCM). This avoids fractions during intermediate steps. Use substitution when one variable already has a coefficient of 1 or -1.

Question 3

What does it mean geometrically when a linear system has no solution or infinite solutions?

Accepted Answer

A 2x2 system represents two straight lines in the coordinate plane. If the lines intersect at one point, there is a unique solution (x, y). If the lines are parallel with equal slopes and different intercepts, there is no solution. If the lines are coincident (identical), there are infinitely many solutions.