Systems with No Solution or Infinitely Many Solutions

When you solve a system of two linear equations, there are three possible outcomes: | Case | Graph | What happens when solving | Example | |---|---|---|---| | One solution | Lines intersect at one point | You find x = a, y = b | Most systems | | No solution | Lines are parallel (never meet) | You get a false statement like 0 = 5 | Same slope, different y-intercept | | Infinitely many | Lines are…

For a system in standard form: [formula] Compare the ratios of the coefficients: | Condition | Result | |---|---| | frac{a_1}{a_2} neq frac{b_1}{b_2} | One solution | | frac{a_1}{a_2} = frac{b_1}{b_2} neq frac{c_1}{c_2} | No solution | | frac{a_1}{a_2} = frac{b_1}{b_2} = frac{c_1}{c_2} | Infinitely many |

Example: How many solutions? [formula]