- Recognize when a linear equation has no solution or infinite solutions
no solution / infinite solutions
When solving a linear equation, sometimes the variable cancels out. What remains tells you the number of solutions:
-
If you reach a true statement like [latex]0=0[/latex], there are infinitely many solutions.
-
If you reach a false statement like [latex]0=5[/latex], there is no solution.
Decide how many solutions each equation has.
-
[latex]3x+6 = 3(x+2)[/latex]
[latex]\begin{align} 3x+6 &= 3(x+2) \\ 3x+6 &= 3x+6 && \text{distribute} \\ 3x+6-3x &= 3x+6-3x && \text{subtract }3x \\ 6 &= 6 && \text{always true} \end{align}[/latex]
Infinitely many solutions.
-
[latex]2x-5 = 2x+1[/latex]
[latex]\begin{align} 2x-5 &= 2x+1 \\ 2x-5-2x &= 2x+1-2x && \text{subtract }2x \\ -5 &= 1 && \text{false} \end{align}[/latex]
No solution.
-
[latex]4x+1 = 5x-3[/latex]
[latex]\begin{align} 4x+1 &= 5x-3 \\ 4x-5x+1 &= 5x -3 -5x && \text{subtract }5x \\ -x+1 &= -3 \\ -x +1 - 1&= -3 - 1 && \text{subtract }1 \\ - x &= - 4 \\ x &= 4 && \text{divide by }-1 \end{align}[/latex]
One solution: [latex]x=4[/latex]