Certainly! Let's carefully convert the given linear programming problem into its standard form.
### Original Problem
Maximize [tex]\( z = -2x_1 \)[/tex]
Subject to:
[tex]\[ 4x_1 - x_2 - 5x_3 = 10 \][/tex]
### Breaking Down the Problem:
1. Objective Function:
The original objective function is:
[tex]\[
z = -2x_1
\][/tex]
Since standard linear programming problems typically require the objective to be maximized without any explicit constraints on solely maximizing a single variable, this part remains unchanged:
[tex]\[
\text{Maximize } z = -2x_1
\][/tex]
2. Equality Constraints:
The constraint given is:
[tex]\[
4x_1 - x_2 - 5x_3 = 10
\][/tex]
This can be reformulated into an inequality form suitable for linear programming standard form:
[tex]\[
4x_1 - x_2 - 5x_3 \leq 10
\][/tex]
To maintain the equality form, we'll also need to account for:
[tex]\[
4x_1 - x_2 - 5x_3 \geq 10
\][/tex]
3. Non-negativity Constraints:
Linear programming problems in standard form require all variables to be non-negative:
[tex]\[
x_1 \geq 0, \quad x_2 \geq 0, \quad x_3 \geq 0
\][/tex]
### Standard Formulation:
After converting the constraints properly, we summarize the LP problem in standard form:
Objective:
[tex]\[
\text{Maximize } z = -2x_1
\][/tex]
Subject to:
1. Inequality form of given constraints:
[tex]\[
4x_1 - x_2 - 5x_3 \leq 10
\][/tex]
2. Non-negativity constraints:
[tex]\[
x_1 \geq 0
\][/tex]
[tex]\[
x_2 \geq 0
\][/tex]
[tex]\[
x_3 \geq 0
\][/tex]
It should be noted that the standard form often includes slack variables to convert inequalities into equalities for standard computational approaches, but the problem as stated doesn't strictly demand these additions explicitly unless preparing for solution algorithms like the Simplex method.
### Final Answer:
The standard form of the linear programming problem is:
Maximize [tex]\( z = -2x_1 \)[/tex]
Subject to:
[tex]\[
4x_1 - x_2 - 5x_3 \leq 10
\][/tex]
[tex]\[
x_1 \geq 0
\][/tex]
[tex]\[
x_2 \geq 0
\][/tex]
[tex]\[
x_3 \geq 0
\][/tex]
In this framework, the problem is now set up correctly to be solved using linear programming techniques.
