Section 1.3: Applications of Linear Equations A good approach to mathematical modeling is a two-stage approach, using a verbal description to form a verbal model and then, after assigning labels to the quantities, forming an algebraic equation.

Write a mathematical model for the problem and solve. The sum of three consecutive natural numbers is 939. Find the numbers. Let 𝑛 = the first natural number Let 𝑛+1= the second natural number Let 𝑛+2= the third natural number So, 𝑛+ 𝑛+1 + 𝑛+2 =939 3𝑛+3=939 3𝑛=936 𝑛= 936 3 =312 The three numbers are 312, 313, 314

Write a mathematical model for the problem and solve. One positive number is 1 4 of another number. The difference between the numbers is 39. Find the numbers. Let 𝑥 be the smaller number. Let 𝑦 be the larger number. So…. 𝑥= 1 4 𝑦 and 𝑦−𝑥=39 Using substitution… 𝑦− 1 4 𝑦=39 or 3 4 𝑦=39 𝑦=39∙ 4 3 =52 52 is the larger number, 13 is the smaller number.

A salesperson’s weekly paycheck is 30% more than a second salesperson’s paycheck. The two paychecks total $920. Find the amount of each paycheck. Let 𝑥= the amount of the first salesperson’s paycheck Let 𝑦= the amount of the second salesperson’s paycheck Since the first salesperson’s check is 30% more than the second salesperson’s check, we have 𝑥=𝑦+0.30𝑦 Also, since the sum is $920, we have 𝑥+𝑦=920 or 𝑦+0.30𝑦 +𝑦=920 Combine like terms and solve: 2.30𝑦=920 → 𝑦=$400 Thus, 𝑥=$520

The price of a swimming pool has been discounted 15.5%. The sale price is $1,233.70. Find the original list price of the pool. Let 𝑥 represent the original list price. The following equation represents the 15.5% discount: 𝑥−0.155𝑥=1233.70 0.845𝑥=1233.70 𝑥= 1233.70 0.845 =$1,460

To get an A in a course, you must have an average of at least 90 on four tests of 100 points each. The scores on your first three tests are 88, 92, and 81. What is the minimum you must score on the fourth test to get an A in the course? Let 𝑥 represent the minimum score needed on the fourth test. To calculate a desired average of 90 for the four exams, we have this equation: 88+92+81+𝑥 4 =90 or 261+𝑥 4 =90 261+𝑥=360 𝑥=99