Add money amounts - word problems

Learning Objectives Translate complex financial word problems into a system of linear inequalities. Define variables to represent quantities and constraints in a money-based scenario. Accurately graph a system of linear inequalities to find a feasible region representing all possible financial outcomes. Identify the vertices of a feasible region by solving systems of linear equations. Formulate an objective function to represent profit, revenue, or cost. Determine the optimal solution (e.g., maximum profit, minimum cost) by evaluating the objective function at the vertices of the feasible region. Ever planned a fundraiser or managed a budget for a project and wondered how to get the most bang for your buck? 💸 Let's use advanced math to solve those exact kinds of puzzle...

TermDefinitionExample System of Linear InequalitiesA set of two or more linear inequalities containing the same variables. The solution to the system is the overlapping region that satisfies all inequalities simultaneously.A student has at most $50 to spend on notebooks (x) and pens (y). Notebooks cost $5 and pens cost $2. They must buy at least 8 items in total. The system is: `5x + 2y ≤ 50` and `x + y ≥ 8`. ConstraintA limitation or restriction in a problem, which is expressed as a linear inequality.A bakery has a maximum of 100 hours of labor available per week. This is a constraint on production. Feasible RegionThe solution set for a system of inequalities, represented graphically as the intersection of the shaded regions for each inequality. Any point (x, y) within this region is a v...

Standard Form of a Linear Inequality Ax + By \leq C \quad \text{or} \quad Ax + By \geq C Used to represent constraints. 'x' and 'y' are the decision variables (e.g., number of items). 'A' and 'B' are the per-unit costs or values. 'C' is the total constraint value (e.g., total budget, total hours). Objective Function for Optimization f(x, y) = ax + by This function represents the quantity you want to optimize. For a profit maximization problem, 'a' and 'b' would be the profit per unit for items 'x' and 'y', respectively. The goal is to find the (x, y) pair in the feasible region that results in the largest or smallest value for this function. Corner Point Principle \text{Maximum or...