Linear programming

Learning Objectives Graph a linear inequality in two variables, correctly identifying the boundary line and shaded region. Identify the solution set, or feasible region, for a system of linear inequalities. Algebraically determine the vertices (corner points) of a feasible region by solving systems of linear equations. Verify if a given ordered pair is a solution to a system of linear inequalities. Model real-world constraints, such as resource limitations, using a system of linear inequalities. Distinguish between bounded and unbounded feasible regions and understand the implications for optimization problems. How can a company decide how many phones and tablets to produce to maximize profit, given limited parts and labor hours? 📱⚙️ This is the core question linear program...

TermDefinitionExample Linear InequalityA mathematical statement that compares two linear expressions using an inequality symbol (<, >, ≤, ≥). Its graph is a half-plane.`3x - 2y ≤ 6` is a linear inequality. All points (x, y) that satisfy this condition are part of its solution. System of Linear InequalitiesA set of two or more linear inequalities that share the same variables. The solution to the system is the set of all points that satisfy every inequality simultaneously.`x ≥ 0`, `y ≥ 0`, `x + y ≤ 10` Feasible RegionThe graphical representation of the solution set for a system of linear inequalities. It is the area where all the shaded regions of the individual inequalities overlap.For the system `x ≥ 0`, `y ≥ 0`, and `x + y ≤ 5`, the feasible region is a triangle in the first quadr...

Graphing a Linear Inequality 1. Graph the boundary line. \n2. Use a solid line for `≤` or `≥`. \n3. Use a dashed line for `<` or `>`. \n4. Pick a test point (e.g., (0,0)) not on the line. \n5. If the test point satisfies the inequality, shade the region containing it. If not, shade the other region. This procedure is used to visually represent all possible solutions for a single linear inequality in two variables. Finding a Vertex To find the vertex at the intersection of two boundary lines, `a_1x + b_1y = c_1` and `a_2x + b_2y = c_2`, solve the system of two linear equations using substitution or elimination. This algebraic method precisely calculates the coordinates of the corner points of the feasible region, which are the candidates for optimal solutions.