More applications of Linear Equations Section 1.7 More applications of Linear Equations

Introduction to Problem Solving Mathematical Modeling- The process of translating phrases or sentences into algebraic expressions or equations. Verbal Model Verbal Description Algebraic Equation

Total interest ($1,110) Interest earned at 8% Interest earned at 7% Example 1: A professor has $15,000 to invest for one year, some at 8% and the rest at 7% . If she wants to earn $1,110 from these investments, how much should she invest at each rate? Total interest ($1,110) Interest earned at 8% Interest earned at 7% VERBAL MODEL i = p • r • t 8% investment 7% investment .08x x .08 1 .07(15,000 – x) 15,000 - x .07 1 0.08x + 0.07( 15,000 – x ) = 1,110 ALGEBRAIC EQUATION

Example 1 (continued) 0.01x = 60 0.08x + 0.07(15,000 – x ) = 1,110 x = 6,000 Multiply both sides by 100 Therefore, $6,000 should be invested at 8% and $9,000 should be invested at 7%.

55 mph 50 mph 157.5 miles Rockford Wausau Example 2: A car leaves Rockford traveling toward Wausau at the rate of 55 mph. At the same time, another car leaves Wausau traveling toward Rockford at the rate of 50 mph. How long will it take them to meet if the cities are 157.5 mile apart? 55 mph 50 mph 157.5 miles Rockford Wausau

Rate x Time = Distance Faster car Slower car 55 t 55t 50 t 50t VERBAL Distance of Distance of MODEL faster car + slower car = Distance between the cities Rate x Time = Distance Faster car Slower car 55 t 55t 50 t 50t 55t + 50t = 157.5 105t = 157.5 t = 1.5 hrs. ALGEBRAIC EQUATION

Amt. fat in 12L of 4% Amt. fat in 1% milk Amt. fat in 2% mixture Example 4: A container is partially filled with 12 liters of whole milk containing 4% butterfat. How much 1% milk must be added to get a mixture that is 2% butterfat? VERBAL MODEL Amt. fat in 12L of 4% Amt. fat in 1% milk Amt. fat in 2% mixture Amt of 4% milk = 12L Amt of 1% milk = x Amt of fat in 4% milk = .04(12) Amt of fat in 1% milk = 0.01x Amt of fat in mixture = 0.02(x + 12) ALGEBRAIC .04(12) + .01x = .02(x + 12) EQUATION

48 + x = 2 (x + 12 ) Multiply both sides by 100 48 + x = 2x + 24 Example 4 (continued) .04(12) + 0.01x = 0.02 (x + 12 ) .48 + .01x = .02 (x + 12) 48 + x = 2 (x + 12 ) Multiply both sides by 100 48 + x = 2x + 24 x = 24 Therefore, 24 liters of 1 % milk should be added to get a mixture that is 2 % butterfat.