7.1 Linear and Nonlinear Systems of Equations Substitution Solving a System of 2 equations in 2 variables. x + y = 4 x - y = 2 Solve for x or y in the easiest way possible and in either equation. From eq. 1 y = 4 - x Substitute 4 - x in for y in equation 2. x - (4 - x) = 2 x - 4 + x = 2 2x = 6 x = 3 and by backsubstituting, y = 1

Note: multiply this equation by 100 so we don’t have any decimals. A total of $12,000 is invested in two funds paying 9% and 11% simple interest. If the yearly interest is $1,180, how much of the $12,000 is invested at each rate? So what are the two equations? x + y = 12,000 two funds totaling $12,000 .09x + .11y = 1,180 Interest formula. RP = I Note: multiply this equation by 100 so we don’t have any decimals. Solving for x in equation 1 we have: x = 12,000 - y Substituting, we have 9(12,000 - y) + 11y = 118,000 108,000 - 9y + 11y = 118,000 2y = 10,000 and y = $5,000 and x = $7,000

Solve: x2 - x - y = 1 -x + y = -1 Solve for y in equation 2 y = x - 1 Now, substitute into eq. 1 x2 - x - (x - 1) = 1 x2 - 2x + 1 = 1 x2 - 2x = 0 Factor x(x - 2) = 0 x = 0, x = 2 Now, we need to back substitute to find our y values. For x = 0, y = -1 For x = 2, y = 1 Note: We must check all answers to make sure that they all work. These do.

A small business invests $10,000 in equipment to produce a product A small business invests $10,000 in equipment to produce a product. Each unit of the product costs $0.65 to produce and is sold for $1.20. How many items must be sold before the business breaks even? The total cost of producing x units is what? C = 0.65x + 10,000 The revenue obtained by selling x units is what? R = 1.2x The break-even point occurs when R = C. Hence, 1.2x = 0.65x + 10,000 .55x = 10,000 and x = 18,182 units