Systems of Equations and Inequalities Chapter 4 Systems of Equations and Inequalities

Chapter Sections 4.1 – Solving Systems of Linear Equations in Two Variables 4.2 – Solving Systems of Linear Equations in Three Variables 4.3 – Systems of Linear Equations: Applications and Problem Solving 4.4 – Solving Systems of Equations Using Matrices 4.5 – Solving Systems of Equations Using Determinants and Cramer’s Rule 4.6 – Solving Systems of Linear Inequalities Chapter 1 Outline

Systems of Linear Equations: Applications and Problem Solving § 4.3 Systems of Linear Equations: Applications and Problem Solving

Use Systems of Equations to Solve Applications Example The combined land area of North Carolina and South Carolina is 78,820 square miles. The difference between the two state’s land areas is 18, 602 square miles. If North Carolina has the larger land area, find the land area of each state.

Use Systems of Equations to Solve Applications Solution Understand We need to determine the land area of North Carolina and the land area of South Carolina. We will use two variables and therefore will need to determine two equations. Translate Let N= the land area of North Carolina Let S = the land area of South Carolina

Use Systems of Equations to Solve Applications Since the total area of the two states is 78, 820 square miles, the first equation is N + S = 78,820 Since North Carolina has a larger land area and since the difference in the land area is 18,602 square miles, the second equation is N – S = 18,602

Use Systems of Equations to Solve Applications The system of two equations is N + S = 78,820 N – S = 18,602 Carry Out We will use the addition method to solve this system of equations. 2N = 97,422 N = 48,711

Use Systems of Equations to Solve Applications Thus, N = 48,711. To determine the value for S, substitute 48,711 into the equation. N + S = 78,820 48,711 + S = 78,820 S = 30,109 Answer The land area of North Carolina is 48,711 square miles and the land area of South Carolina is 30,109 square miles.

Using Tables Example Chung Song, a chemist with Johnson and Johnson, wishes to create a new household cleaner containing 30% trisodium phosphate (TSP). Chung needs to mix a 16% TSP solution with a 72% TSP solution to get 6 liters of 30% TSP solution. How many liters of the 16% solution and of the 72% solution will he need to mix? Use a table to organize your information.

Using Tables Let x = number of liters of the 16% solution Let y = number of liters of the 72% solution

Using Tables Solution Strength of Solution Number of Liters Amount of TSP 16% solution 0.16 x 0.16x 72% solution 0.72 y 0.72y Mixture 0.30 6 0.30(6)

Using Tables Since the sum of the volumes of the 16% solution and 72% solution is 6 liters, our first equation is x + y = 6 The second equation comes form the fact that the solutions are mixed. Amount of TSP in 16% solution Amount of TSP in 72% solution Amount of TSP in mixture + = Therefore, the systems of equation is x + y = 6 0.16x + 0.72y =- 0.30(6)

Using Tables Carry Out Solving x + 6 = 6 for y, we get y = -x + 6. Substituting –x + 6 for y in the second equation gives us Therefore, Chung must use 4.5 liters of the 16% solution. Since the two solutions must total 6 liters, he must use 6 – 4.5 or 1.5 liters of the 72% solution

Three Variables Tiny Tots Toys must borrow $25,000 to pay for an expansion. It is not able to obtain a loan for the total amount from a single bank, so it takes out loans from three different banks. It borrows some of the money at a bank that charges 8% interest. At the second bank, it borrows $2,000 more than one-half the amount borrowed form the first bank. The interest rate at the second bank is 10%. The balance of the $25,000 is borrowed from a third bank, where Tiny Tots pays 9% interest. The total annual interest Tiny Tots Toys pays for the three loans is $2220. How much does it borrow at each rate?

Three Variables Let x = amount borrowed at first bank Let y = amount borrowed at second bank Let z = amount borrowed at third bank Since the total amount borrowed is $25,000 we know that x + y + z = 25,000

Three Variables At the second bank, Tiny Tots borrows $2000 more than one-half the amount borrowed from the first bank. Therefore, our second equation is Our last equation comes from the fact that the total annual interest charged by the three banks is $2220. The interest at each bank is found by multiplying the interest rate by the amount borrowed.

Three Variables Thus, our system of equations is Both sides of equation 2 can be multiplied by 2 to remove fractions

Three Variables The decimals in equation 3 can be removed by multiplying both sides of the equation by 100. This gives 8x +10y + 9z = 222,000 Our simplified system of equations is therefore

Three Variables Carry Out There are various ways of solving this system. Let’s use equation 1 and 3 to eliminate the variable z. -9x - 9y - 9z = -225,000 8x + 10y + 9z = 222,000 -x + y = -3,000 Now we use equations 2 and 4 to eliminate the variable x and solve for y. x - 2y = -4,000 -x + y = -3,000 -y = -7,000 y = 7,000

Three Variables Now that we know the value of y we can solve for x. Now we solve for z. Answer Tiny Tots Toys borrows $10,000 at 8%, $7000 at 10%, and $8000 at 9% interest.