Answer :
To convert an inequality into an equation using a slack variable, follow these steps:
1. Understand the Requirement:
- You have the inequality: [tex]\(7x_1 + 8x_2 + 7x_3 \leq 145\)[/tex].
- To convert this inequality into an equation, we need to account for the "less than or equal to" part by adding a non-negative slack variable.
2. Add the Slack Variable:
- A slack variable is added to the left side of the inequality. Its purpose is to "take up the slack" or difference between the left and right sides when the inequality is less than the maximum allowable (here 145).
- Let's call the slack variable [tex]\(s_1\)[/tex].
- The expression becomes: [tex]\(7x_1 + 8x_2 + 7x_3 + s_1 = 145\)[/tex].
3. Ensure Non-negativity of the Slack Variable:
- The slack variable [tex]\(s_1\)[/tex] must be greater than or equal to 0, since it represents any "unused" portion between the left side of the original inequality and 145.
4. Choose the Correct Option:
- After adding the slack variable, the inequality is transformed into the equation: [tex]\(7x_1 + 8x_2 + 7x_3 + s_1 = 145\)[/tex].
- This corresponds to the option B.
Thus, the correct corresponding equation after adding the slack variable is:
[tex]\[7x_1 + 8x_2 + 7x_3 + s_1 = 145\][/tex]
Option B is the correct choice.
1. Understand the Requirement:
- You have the inequality: [tex]\(7x_1 + 8x_2 + 7x_3 \leq 145\)[/tex].
- To convert this inequality into an equation, we need to account for the "less than or equal to" part by adding a non-negative slack variable.
2. Add the Slack Variable:
- A slack variable is added to the left side of the inequality. Its purpose is to "take up the slack" or difference between the left and right sides when the inequality is less than the maximum allowable (here 145).
- Let's call the slack variable [tex]\(s_1\)[/tex].
- The expression becomes: [tex]\(7x_1 + 8x_2 + 7x_3 + s_1 = 145\)[/tex].
3. Ensure Non-negativity of the Slack Variable:
- The slack variable [tex]\(s_1\)[/tex] must be greater than or equal to 0, since it represents any "unused" portion between the left side of the original inequality and 145.
4. Choose the Correct Option:
- After adding the slack variable, the inequality is transformed into the equation: [tex]\(7x_1 + 8x_2 + 7x_3 + s_1 = 145\)[/tex].
- This corresponds to the option B.
Thus, the correct corresponding equation after adding the slack variable is:
[tex]\[7x_1 + 8x_2 + 7x_3 + s_1 = 145\][/tex]
Option B is the correct choice.
