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10.3 Solving Linear Systems Elimination Method #3 All 3 Methods Graphing Substitution Elimination Using addition/subtraction Using multiplication Solutions One solution No solution Infinitely many solutions Elimination ● We will try to eliminate one variable by adding, subtracting, or multiplying the variable(s) until the two terms are additive inverses. ● We will then add the two equations, giving us one equation with one variable. ● Solve for that variable. ● Then insert the value into one of the original equations to find the other variable Elimination ● The key to solving a system by elimination is getting rid of one variable. ● Let’s review the Additive Inverse Property. ● What is the Additive Inverse of: 3x? -5y? 8p? q? -3x 5y -8p -q ● What happens if we add two additive inverses? We get zero. The terms cancel. Elimination ● Solve the system: ● Notice that the n terms in both equations are additive inverses. So if we add the equations the n terms will cancel. ● So let’s add & solve: ● ● m+n=6 m -n=5 m+n=6 + m- n=5 2m + 0 = 11 2m = 11 m = 11/2 or 5.5 Insert the value of m to find n: 5.5 + n = 6 n = .5 The solution is (5.5, .5). Elimination ● Solve the system: 3s - 2t = 10 4s + t = 6 ● We could multiply the second equation by 2 and the t terms would be inverses OR We could multiply the first equation by 4 and the second equation by -3 to make the s terms inverses. ● Let’s multiply the second equation by 2 to eliminate t. (It’s easier.) 3s - 2t = 10 3s – 2t = 10 2(4s + t = 6) 8s + 2t = 12 Add and solve: 11s + 0t = 22 11s = 22 s=2 ● ● ● Insert the value of s to find the value of t The solution is (2, -2). 3(2) - 2t = 10 t = -2 Elimination Solve the system by elimination: 1. -4x + y = -12 4x + 2y = 6 2. 5x + 2y = 12 -6x -2y = -14 3. 5x + 4y = 12 7x - 6y = 40 4. 5m + 2n = -8 4m +3n = 2