1. **State the problem:** We want to understand how to determine if an equation has no solution or infinitely many solutions. 2. **Recall the types of solutions for linear equations:** - An equation has **no solution** if it simplifies to a contradiction (like $a = b$ where $a \neq b$). - An equation has **infinitely many solutions** if it simplifies to an identity (like $a = a$). 3. **Analyze examples:** - For $t + 1 = t + 1$, subtract $t$ from both sides: $$\cancel{t} + 1 = \cancel{t} + 1$$ $$1 = 1$$ This is always true, so **infinitely many solutions**. - For $8(t + 1) = 8t - 8$, expand left side: $$8t + 8 = 8t - 8$$ Subtract $8t$ from both sides: $$\cancel{8t} + 8 = \cancel{8t} - 8$$ $$8 = -8$$ This is false, so **no solution**. - For $2t = 8t$, subtract $2t$ from both sides: $$\cancel{2t} = 6t$$ $$0 = 6t$$ This is true only if $t=0$, so **one solution**. - For $t = t + 2$, subtract $t$ from both sides: $$\cancel{t} = \cancel{t} + 2$$ $$0 = 2$$ False, so **no solution**. - For $12 - t = t - 12$, add $t$ to both sides and add $12$ to both sides: $$12 = 2t - 12$$ $$12 + 12 = 2t$$ $$24 = 2t$$ $$t = 12$$ One solution. - For $2t + 6 = 2(t + 3)$, expand right side: $$2t + 6 = 2t + 6$$ Subtract $2t + 6$ from both sides: $$\cancel{2t} + \cancel{6} = \cancel{2t} + \cancel{6}$$ $$0 = 0$$ Always true, so **infinitely many solutions**. 4. **Summary:** - **No solution:** When simplification leads to a false statement like $0 = 2$. - **Infinitely many solutions:** When simplification leads to a true statement like $0 = 0$. This is how you can tell whether an equation has no solution or infinitely many solutions.