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Solving Systems of Equations Substitution Method Solving Systems by Substitution {Step 1} Solve for one variable in at least one equation if necessary {Step 2} Substitute the resulting expression into the other equation. {Step 3} Solve the equation equation to obtain the first variable {Step 4} Substitute that value into one of the original equations and solve for the other variable {Step 5} Write the values from Step 3 and Step 4 as an ordered pair, (x, y). {Step 6} Check your solution Example 1 EQ1: y 2 x EQ 2 : y x 5 Both equations are solved for y, so we can substitute either equation into the other. Let’s choose to substitute 2x for y in EQ2. Now y = x + 5 becomes 2x = x + 5 Let’s solve for x ! 2x x 5 x x x5 Plug into EQ1 EQ1: y = 2x becomes y = 2(5), therefore y = 10. The solution point is (5, 10). Example 2 EQ1: 2 x y 5 EQ 2 : y x 4 The second equation is solved for y, so we can substitute x – 4 into y for EQ1. Now 2x + y = 5 becomes 2x + (x – 4) = 5 Simplify Let’s solve for x ! 2 x ( x 4) 5 3x 4 5 4 4 3x 9 x3 Divide both sides by 3 Plug into EQ 2 EQ2: y = x - 4 becomes y = 3 – 4, therefore y = -1. The solution point is (3, -1). Consumer Economics Application One high-speed internet service provider has a $50 setup fee, and costs $30 per month. Another provider has no setup fee and costs $40 per month. After how many months will both providers have the same cost? What will the cost be? If you plan to cancel in one year, which is the cheaper provider? Explain. Write a System of Equations Write an equation for each option. Let y represent the total amount paid and let x represent the number of months. Equation 1: y = 30x + 50 Equation 2: y = 40x Solve the System EQ1 : y 30 x 50 Both equations are solved for y, so we can substitute either equation into the other. EQ2 : y 40 x Let’s choose to substitute 40x for y in EQ1. Now y = 30x + 50 becomes 40x = 30x + 50 Let’s solve for x ! 40 x 30 x 50 30 x 30 x 10 x 50 x5 Divide both sides by 10 Plug into EQ2 EQ2: y = 40x becomes y = 40(5), therefore y = 200. The solution point is (5, 200). Interpret the Results Since x represents the number of months and x = 5, this means that at 5 months both plans will be have equal cost. Since y represents the total cost and y = 200, this means that after 5 months both plans will have cost $500 Which plan is cheaper after 1 year? Plug 12 in for x and compare the results. When x = 12 EQ1 = $410 EQ2 = $480