Question

complete the equation so it has exactly one solution.
$1 + 3m + m = $
$4m$
$4m + 1$
$5m$

Explanation:

Step1: Simplify left side

Simplify \(1 + 3m + m\). Combine like terms: \(3m + m = 4m\), so left side is \(1 + 4m\) (or \(4m + 1\)).

Step2: Analyze each option

• Option 1: \(4m\). Equation: \(4m + 1 = 4m\). Subtract \(4m\): \(1 = 0\) (no solution).
• Option 2: \(4m + 1\). Equation: \(4m + 1 = 4m + 1\). This is an identity (infinitely many solutions). Wait, maybe I made a mistake. Wait, the original left side is \(1 + 3m + m = 4m + 1\). Wait, the problem says "exactly one solution". Wait, maybe I miscalculated. Wait, let's re - evaluate. Wait, no, wait: Wait, if we choose \(5m\):

Equation: \(4m + 1 = 5m\). Subtract \(4m\): \(1 = m\), so \(m = 1\) (exactly one solution). Wait, earlier analysis was wrong. Let's redo:

Left side: \(1+3m + m=4m + 1\).

Option 1: \(4m\): \(4m + 1=4m\) → \(1 = 0\) (no solution).

Option 2: \(4m + 1\): \(4m + 1=4m + 1\) → \(0 = 0\) (infinitely many solutions).

Option 3: \(5m\): \(4m+1 = 5m\) → \(5m-4m=1\) → \(m = 1\) (exactly one solution).

Ah, I see, my initial mistake was in the second option analysis. So to have exactly one solution, the right - hand side should not be equivalent to the left - hand side (for infinite solutions) and should not lead to a contradiction (no solution). So the correct option is \(5m\).

Answer:

\(5m\) (the option with text "5m")