VIEWS: 9 PAGES: 13 POSTED ON: 10/13/2011
MATH 048 SIMULTANEOUS EQUATIONS Solving Systems of Equations Algebraically- Elimination Method • Solving a system of linear equations algebraically using ELIMINATION with addition and subtraction is easiest when the equations are in standard form(i.e. x and y on both sides and constant on other side). Solving a system of equations by elimination using addition and subtraction. Step 1: Put the equations in Standard Form: Ax + By = C Standard Form. Look for variables that have the Step 2: Determine which same coefficient. If none then multiply one or variable to eliminate. both equations by an appropriate number to make same or opposites . Step 3: Add or subtract the Solve for the variable. equations. Step 4: Plug back in to find the Substitute the value of the variable other variable. into the equation. Substitute your ordered pair into Step 5: Check your solution. BOTH equations. . 1) Solve the system using elimination x+y=5 3x – y = 7 Step 1: Put the equations in They already are! Standard Form. The y’s have the same Step 2: Determine which coefficient. variable to eliminate. Add to eliminate y. x+ y=5 Step 3: Add or subtract the (+) 3x – y = 7 equations. 4x = 12 x=3 1) Solve the system using elimination contd. x+y=5 3x – y = 7 x+y=5 Step 4: Plug back in to find the other variable. (3) + y = 5 y=2 (3, 2) Step 5: Check your solution. (3) + (2) = 5 3(3) - (2) = 7 The solution is (3, 2). What do you think the answer would be if you solved using substitution? Examples. Solve using Elimination a) x y 9 b) x 2 y 8 c) x 2 y 9 x- y 3 x-2y 4 2 x 18 4 y d ) m 2n 14 e) 4 y 3x -1 f ) 9 x 4 y -25 3n m 18 6 y - 7 x -36 6 x -14 y -25 g) 2 3 4 x y h) 0.04 x 0.05 y 44 i) 2( x y) y 1 x 4 3 4 y x y 1000 3( x 1) y -3 Solving Systems of Equations • These notes show how to solve a system algebraically using SUBSTITUTION. • The method is easiest to use when the system is a system of linear equations where one of the variable’s coefficient is 1. Solving a system of equations by substitution Pick the easier equation. For mixed Step 1: Solve an equation for solving the linear equation for one variable. variable is usually easier. Put the equation solved in Step 1 Step 2: Substitute into the other equation. Step 3: Solve the equation. Get the variable by itself. Step 4: Plug back in to find the Substitute the value of the variable other variable. into the equation. Substitute your ordered pair into Step 5: Check your solution. BOTH equations. 1) Solve the system using substitution x+y=5 y=3+x Step 1: Solve an equation for The second equation is one variable. already solved for y! x+y=5 Step 2: Substitute x + (3 + x) = 5 2x + 3 = 5 Step 3: Solve the equation. 2x = 2 x=1 1) Solve the system using substitution x+y=5 y=3+x x+y=5 Step 4: Plug back in to find the other variable. (1) + y = 5 y=4 (1, 4) Step 5: Check your solution. (1) + (4) = 5 (4) = 3 + (1) The solution is (1, 4). What do you think the answer would be if you graphed the two equations? Examples. Solve using substitution method a) y 3x b) x 4 y c) a b 3 x y 83 x 8 y 52 a 3b 1 d ) 4x y 0 8x 1 y 7 4 More Examples • Solve the following using any method. a ) 3 p 2r 7 b) y x p 2r 11 3 y 3x 4 Examples - Applications of Systems • 1) The sum of two numbers is 56. Twice the smaller number exceeds the larger number by 22. Find the numbers. • 2)The sum of two numbers is 50. If twice the larger is subtracted from 4 times the smaller, the result is 8. Find the numbers. • 3) Mrs. B bought 3 cans of corn and 5 cans of tomatoes for $1.82. The following week, she bought2 cans of corn and 3 cans of tomatoes for $1.11,paying the same prices. Find the cost of a can of corn and the cost of a can of tomatoes.