Solving Systems of Equations
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- Horatio Palmer
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1 Solving Systems of Equations
2 Solving Systems of Equations What are systems of equations? Two or more equations that have the same variable(s)
3 Solving Systems of Equations There are three ways to solve systems of equations Solve by graphing Solve by elimination Solve by substitution
4 Solving Equations To solve systems of equations, you must first be able to solve equations in one variable. 3x + 5 x = x First get all variables on one side and constants on the other. 3x + 5 x = x - 5-5x +5-5x Add/subtract 3x -6x = 18 Combine like terms -3x = 18 divide both sides by 3 x = 6
5 Solving Equations with 2 Variables 1. combine like terms 2. move variables so that the variable you want to solve for is on one side, everything else is on the other 3x + 2y = 3 3x = -2y + 3 X = -2y/3 + 1 X = ⅔y + 1
6 Solve this system of equations using the addition or subtraction (elimination) method. Check. x 2y = 14 x + 3y = 9
7 Simultaneous equations got you baffled? Relax! You can do it! Think of the adding or subtracting method as temporarily "eliminating" one of the variables to make your life easier. Systems of Equations may also be referred to as "simultaneous equations". "Simultaneous" means being solved "at the same time".
8 Solve this system of equations and check: x -2y = 14 x + 3y = 9 First, be sure that the variables are "lined up" under one another. In this problem, they are already "lined up". x -2y = 14 x + 3y = 9
9 Decide which variable ("x" or "y") will be easier to eliminate. In order to eliminate a variable, the numbers in front of them (the coefficients) must be the same or negatives of one another. Looks like "x" is the easier variable to eliminate in this problem since the x's already have the same coefficients. x -2y = 14 x + 3y = 9
10 Now, in this problem we need to subtract to eliminate the "x" variable. Subtract ALL of the sets of lined up terms. (Remember: when you subtract signed numbers, you change the signs and follow the rules for adding signed numbers.) x -2y = 14 -x - 3y = - 9-5y = 5
11 Solve this simple equation. -5y = 5 y = -1
12 Plug "y = -1" into either of the ORIGINAL equations to get the value for "x". x -2y = 14 x - 2(-1) = 14 x + 2 = 14 x = 12
13 Check: substitute x = 12 and y = -1 into BOTH ORIGINAL equations. If these answers are correct, BOTH equations will be TRUE! x -2y = (-1) = = = 14 (check!) x + 3y = (-1) = = 9 9 = 9 (check!)
14 Solve this system of equations and check: 4x + 3y = -1 5x + 4y = 1 You can probably see the dilemma with this problem right away. Neither of the variables have the same (or negative) coefficients to eliminate. Yeek! 4x + 3y = -1 5x + 4y = 1
15 In this type of situation, we must MAKE the coefficients the same (or negatives) by multiplication. You can MAKE either the "x" or the "y" coefficients the same. Pick the easier numbers. In this problem, the "y" variables will be changed to the same coefficient by multiplying the top equation by 4 and the bottom equation by 3.
16 Remember: * you can multiply the two differing coefficients to obtain the new coefficient if you cannot think of another smaller value that will work. * multiply EVERY element in each equation by your adjustment numbers. 4(4x + 3y = -1) 3(5x + 4y = 1) 16x + 12y = -4 15x + 12y = 3
17 Now, in this problem we need to subtract to eliminate the "y" variable. (Remember: when you subtract signed numbers, you change the signs and follow the rules for adding signed numbers.) 16x + 12y = -4-15x -12y = - 3 x = - 7
18 Plug "x = -7" into either of the ORIGINAL equations to get the value for "y". 5x + 4y = 1 5(-7) + 4y = y = 1 4y = 36 y = 9 X = -7; y = 9
19 Does it check? 4x + 3y = -1 5x + 4y = 1
20 Solve this system of equations and check: 4x - y = 10 2x = 12-3y
21 Solving equations with the substitution method. The substitution method is used to eliminate one of the variables by replacement when solving a system of equations. Think of it as "grabbing" what one variable equals from one equation and "plugging" it into the other equation
22 Solve this system of equations (and check): 3y -2x = 11 y + 2x = 9 Solve one of the equations for either "x =" or "y =". This example solves the second equation for "y =". 3y -2x = 11 y = 9-2x
23 Replace the "y" value in the first equation by what "y" now equals. Grab the "y" value and plug it into the other equation. 3(9-2x) - 2x = 11
24 Solve this new equation for "x". (27-6x) - 2x = x - 2x = x = 11-8x = -16 x = 2
25 Place this new "x" value into either of the ORIGINAL equations in order to solve for "y". Pick the easier one to work with! y + 2x = 9 or y = 9-2x y = 9-2(2) y = 9-4 y = 5
26 Check: substitute x = 2 and y = 5 into BOTH ORIGINAL equations. If these answers are correct, BOTH equations will be TRUE! 3y -2x = 11 3(5) - 2(2) = = = 11 (check!) y + 2x = (2) = = 9 9 = 9 (check!)
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Stat330 - Solution to Homework 2 1 Kolmogorov (a For any two evens, B Ω, Because and P ( B = P ( P ( B. = ( B ( B, ( B ( B =, axiom iii implies that P ( = P ( B + P ( B. The statement follows from subtraction.