Solve pairs of linear equations graphically and algebraically, and determine whether a system has one, no, or infinite solutions.
For a₁x+b₁y=c₁ and a₂x+b₂y=c₂: if a₁/a₂≠b₁/b₂, unique solution; if a₁/a₂=b₁/b₂≠c₁/c₂, no solution; if all three ratios equal, infinite solutions.
Check the consistency of: 2x+3y=5, 4x+6y=10
Ratios: 2/4=3/6=5/10=1/2; all equal, so infinite solutions
Check the consistency of: x+y=4, x+y=7
Ratios: 1/1=1/1, but 4≠7; parallel lines, no solution
Substitution and elimination methods solve pairs of linear equations to find a unique intersection point when one exists.
Solve: x+y=10, 2x-y=5
Adding: 3x=15, x=5; y=5
Solve: 3x+2y=12, x-y=1
From 2nd: x=y+1; sub: 3(y+1)+2y=12; 5y=9; y=9/5, x=14/5
1. Check the consistency of: x+2y=6, 3x+6y=18
2. Check the consistency of: 2x+y=5, 2x+y=9
3. Solve: x+y=12, x-y=4
4. Solve: 2x+3y=13, x-y=1
5. For what value of k does 2x+3y=7 and kx+6y=14 have infinite solutions?