Questions: Write the associated linear equation for the inequality x₁+x₂ ≤ 8. Use s₁ as the slack variable.

Transcript text: Write the associated linear equation for the inequality $\mathrm{x}_{1}+\mathrm{x}_{2} \leq 8$. Use $\mathrm{s}_{1}$ as the slack variable.

Solution

Solution Steps

To convert the inequality $ x_1 + x_2 \leq 8 $ into an equation, we introduce a slack variable $ s_1 $ to account for the difference between the left and right sides of the inequality. The slack variable is added to the left side to make it equal to the right side, resulting in the equation $ x_1 + x_2 + s_1 = 8 $.

Step 1: Define the Inequality

We start with the inequality given in the problem: \[ x_1 + x_2 \leq 8 \]

Step 2: Introduce the Slack Variable

To convert the inequality into an equation, we introduce a slack variable $ s_1 $. This variable accounts for the difference between the left and right sides of the inequality. Thus, we rewrite the inequality as: \[ x_1 + x_2 + s_1 = 8 \]

Step 3: Final Equation

The final equation, incorporating the slack variable, is: \[ s_1 + x_1 + x_2 = 8 \]