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The steps to solve the equation 3 x plus left-parenthesis 10 plus 2 x right-parenthesis equals 0 are shown.



Drag and drop the properties of equality to the table to justify each step.

Response area with 3 blank spaces
Solution Step

Justification

3 x plus left-parenthesis 10 minus 2 x right-parenthesis equals 0
Original equation

3 x plus left-parenthesis negative 2 x plus 10 right-parenthesis equals 0
Blank space 1 empty

left-parenthesis 3 x minus 2 x right-parenthesis plus 10 equals 0
Blank space 2 empty

x plus 10 equals 0
Simplification

x plus 10 plus left-parenthesis negative 10 right-parenthesis equals 0 plus left-parenthesis negative 10 right-parenthesis
Blank space 3 empty

x equals negative 10
Simplification


Answer options with 6 options.

1 answer

To justify each step in solving the equation \(3x + (10 + 2x) = 0\), we can use the properties of equality. Below are the justifications for the statements provided in the table:

  1. Original equation:

    • Justification: Original equation (This matches the statement as it's the starting equation).

  2. Blank space 1 (3x + (10 - 2x) = 0 to 3x + (-2x + 10) = 0):

    • Justification: Associative Property of Addition (This allows us to rearrange terms within the parentheses without changing the value).

  3. Blank space 2 (From (3x - 2x) + 10 = 0 to x + 10 = 0):

    • Justification: Combine Like Terms (This step simplifies \(3x - 2x\) to \(x\)).

  4. Blank space 3 (x + 10 + (-10) = 0 + (-10)):

    • Justification: Addition Property of Equality (This states that you can add the same value to both sides of an equation without changing the equality).

Here’s how the filled table looks:

| Solution Step | Justification | |--------------------------------------------------------|----------------------------------------| | \(3x + (10 + 2x) = 0\) | Original equation | | \(3x + (-2x + 10) = 0\) | Associative Property of Addition | | \((3x - 2x) + 10 = 0\) | Combine Like Terms | | \(x + 10 + (-10) = 0 + (-10)\) | Addition Property of Equality | | \(x = -10\) | Simplification |

This setup provides a clear logical sequence for each step taken in solving the equation.