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Author: Thomas Geisler Last updated: June 24, 2013 Unit 3 Solving Linear Equation Word Problems Avery Point Academic Center Unit Three: Linear Equations - 1 Relevant terms and formulas: Two-variables linear equations are of the form: ax + by = c or ax −by = c One-variable linear equations are of the form: ay = b or cx = d where x and y are variables and a, b, c, and d are constants. Unit Three: Linear Equations - 2 The standard forms of 2-variable and 1-variable linear equations are as follows: Formula 1: y = mx + b (2-variables) Formula 2: y = c or x = d (1-variable) (The graph of a linear equation is a straight line.) For a linear equation in standard form: The slope of a linear equation is the coefficient of x, m, and indicates how many units y changes (positively or negatively) for every one unit increase in x. Unit Three: Linear Equations - 3 Formula 3: If two points (x1, y1) and (x2, y2) satisfy a linear equation, the slope of the equation is given by the formula: m = y2 − y1 x2 − x1 Formula 4: The Point Slope Formula: A linear equation may be determined if one point on the equation, (x1, y1), and its slope, m, are known, using the following equation: y − y1 = m(x − x1) Unit Three: Linear Equations - 4 Rule: The solution of a pair of linear equations are the values of x and of y which are consistent with both equations. For most pairs, there is a unique solution, i.e., a single value of x and of y [x1 and y1, also written (x1, y1)], which satisfies both equations. (NOTE: In some cases, there is no solution; in other cases, there are infinite solutions.) Unit Three: Linear Equations - 5 Solution Method: The four steps to solve a pair of linear equations: 1. Put both equations into standard form, if they are not already: y = mx + b and y = nx + c. 2. Equate the two right-hand sides of the equations to form a single equation in the single variable x: mx + b = nx + c. (continued . . .) Unit Three: Linear Equations - 6 (continued. . .) 3. Solve for x. 4. Substitute that value of x into either of the two standard form equations and solve for y. Solution: The values of x and y you have calculated are the solutions to the problem. Unit Three: Linear Equations - 6 Many word problems may be reduced to linear equation form. Sometimes, the linear equation or equations are explicitly stated in the problem. In other cases, you must construct the linear equation(s) from the data given in the problem. If you are given sufficient data in the problem, such as the x and y values of two points satisfying an equation or the x and y values of one point and the equation’s slope, you can create the equation. Unit Three: Word Problem 1 - I Problem 1: In a month, a widget company knows that it has $20,000 in fixed costs before any widget is produced and that each widget produced during that month costs $5 to make in addition to the fixed costs. (a) Devise a formula for the company’s total cost, C, in a month when n widgets are made. (b) Use the formula to determine the company’s total cost of making 55,000 widgets in that month. What’s a Widget? “Widget” is a name in economics for an unidentified product in problems where the specific product produced is irrelevant to the problem. It came from a Broadway comedy of the 1920s. But wouldn’t you know, now ‘widgets’ are host software systems for physically inspired applets on computers (see below). Unit Three: Word Problem 1 - II Solution: (a) Solve the first part. 1. The first step is to read and reread the problem to make sure we understand it. 2. We are given the following data: Fixed costs of $20,000 are incurred even if there are no widgets made. In addition to the fixed costs, the actual production of a each widget is $5 per widget. Unit Three: Word Problem 1 - III 3. We are asked in (a) to find a formula for the company’s total monthly costs when a given number of widgets are made. What are the variables? There appear to be two variables. The first variable is the number of widgets made in a month, which we designate as n. The second variable is the total cost of making n widgets in a month, which we designate C. (We use n and C since those variables were stated in the problem, but n and C conveniently remind us they stand for number and Cost.) Unit Three: Word Problem 1 - IV 4. Since we have been studying linear equations, the mathematical formula being sought should be a linear equation, preferably in standard form: . y = mx + b We must rewrite the problem in the form of a linear equation, with the given data and the variables we have chosen. The company’s costs include both the fixed costs and the specific costs of making the widgets. Unit Three: Word Problem 1-V The cost of making one widget is $5, so the cost of making n widgets (excluding fixed costs) is: 5n But we must add the fixed costs, $20,000, to 5n, to represent the total costs per month. That means that the formula is: C = 5n + 20,000 which is a linear equation, a formula that is the solution to (a). Since (a) only asked us to create the proper formula, we have now answered (a). Unit Three: Word Problem 1 -VI (b) The formula developed in (a) is used to solve (b): what is the company’s total cost of making 55,000 widgets? 55,000 widgets is the new data in Step 2 in (b)? We created the formula in (a) and are thus ready for Step 6: 5. Substitute 55,000 for n in the formula: C = 5n + 20,000 C = 5(55,000) + 20,000 Unit Three: Word Problem 1 -VII 6. Solve mathematically: C = 275,000 + 20,000 C = 295,000 7. Each term is in dollars; (5 dollars/widget) times 55,000 widgets yields a product in dollars. The answer to (b) is $295,000.