Question
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.
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.
Answers
GPT-4o mini
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.
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.