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6/18/2015 When you solve a system of equations, you’re finding the point where two lines intersect. It’s possible to do this by actually graphing the lines, but there are some problems. It’s time-consuming. What if the lines don’t intersect at whole numbers? You need graph paper. You have to graph very neatly. Graphing is usually just about the least efficient way to solve systems of equations. Fortunately there are other ways to do it. Often the easiest way to solve systems of equations is substitution. 1 6/18/2015 Both in math and in life, substitution means replacing one thing with another. In algebra, substitution means replacing a variable with something it is equal to. Steps for Substitution 1. Solve one of the equations for one variable in terms of the other. (Make it say x = … or y = …) 2 6/18/2015 2. Substitute into the other equation. (Copy the equation, but change one variable to what you just said it equaled.) Substitution is usually an efficient method if at least one of the variables doesn’t have a coefficient. 3x + y = 14 2x + 3y = 0 3x + 5y = 2 x – 3y = 10 3. Solve. 4. Substitute back (to #1) to find the other answer. Use substitution to solve x + 4y = 11 3x – 2y = 12 x + 4y = 11 3x – 2y = 12 x + 4y = 11 3x – 2y = 12 First, solve the top equation for x in terms of y. First, solve the top equation for x in terms of y. x = 11 – 4y 3 6/18/2015 x + 4y = 11 … x = 11 – 4y 3x – 2y = 12 Now, substitute for “x” in the bottom equation. x + 4y = 11 … x = 11 – 4y 3x – 2y = 12 x + 4y = 11 … x = 11 – 4y 3x – 2y = 12 Now, substitute for “x” in the bottom equation. • Change “x” to “11 – 4y” Solve for “y”. 3(11 – 4y) – 2y = 12 Now, substitute for “x” in the bottom equation. • Change “x” to “11 – 4y” 3(11 – 4y) – 2y = 12 Solve for “y”. 3(11 – 4y) – 2y = 12 33 – 12y – 2y = 12 33 – 14y = 12 -14y = -21 y = 1.5 We know … • y = 1.5 • x = 11 – 4y Now find x. 4 6/18/2015 We know … • y = 1.5 • x = 11 – 4y Now find x. x = 11 – 4(1.5) x = 11 – 6 x=5 x + 4y = 11 3x – 2y = 12 The answer is ( 5, 1.5 ) 3x + y = 14 y = 14 – 3x Solve 2x + 3y = 0 2x + 3(14 – 3x) = 0 2x + 42 – 9x = 0 -7x + 42 = 0 -7x = -42 x=6 3x + y = 14 2x + 3y = 0 y = 14 – 3x x=6 y = 14 – 3(6) y = 14 – 18 y = -4 Solve 3x + 5y = 2 x – 3y = 10 So the answer is ( 6, -4 ). 5 6/18/2015 3x + 5y = 2 x – 3y = 10 x = 10 + 3y x = 10 + 3y y = -2 x = 10 + 3(-2) x = 10 – 6 x=4 3(10 + 3y) + 5y = 2 30 + 9y + 5y = 2 14y + 30 = 2 14y = -28 y = -2 Solve ( 4, -2 ) Solve y = 2x + 1 y = 6x – 5 y = 2x + 1 y = 6x – 5 Just change the “y” in one equation to the expression in the other. 6x – 5 = 2x + 1 6x – 5 = 2x + 1 4x – 5 = 1 4x =6 x = 1.5 6x – 5 = 2x + 1 4x – 5 = 1 4x =6 x = 1.5 Since y = 2x + 1 y = 2(1.5) + 1 y=4 … ( 1.5 , 4 ) 6 6/18/2015 Here’s a minor variation on yesterday’s problem … Two tacos and a burrito cost $7.02. One taco and two burritos cost $9.33. How much is a taco, and how much is a burrito? 2t + b = 7.02 t + 2b = 9.33 We can either get “b” by itself in the top equation or “t” by itself in the bottom equation. b = 7.02 – 2t b = 7.02 – 2(1.57) b = 7.02 – 3.14 b = 3.88 So … tacos cost $1.57, and burritos cost $3.88. 2t + b = 7.02 t + 2b = 9.33 It’s easy to solve this using substitution. 2t + b = 7.02 b = 7.02 – 2t t + 2b = 9.33 t + 2(7.02 – 2t) = 9.33 t + 14.04 – 4t = 9.33 -3t + 14.04 = 9.33 -3t = -4.71 t = 1.57 If you did it the other way … 2t + b = 7.02 t + 2b = 9.33 t = 9.33 – 2b 2(9.33 – 2b) + b = 7.02 18.66 – 4b + b = 7.02 18.66 – 3b = 7.02 -3b = -11.64 b = 3.88 7 6/18/2015 t = 9.33 – 2b t = 9.33 – 2(3.88) t = 1.57 Again … burritos are $3.88, and tacos are $1.57. You have some quarters and some nickels. You have 41 coins that are worth $5.45. How many of each type of coin do you have? In problems like this, write 2 equations: • • Another common type of story problem is the mixture problem. Here’s a typical example … In problems like this, write 2 equations: • Number • Value n + q = 41 5n + 25q = 545 n = 41 – q Number n + q = 41 Value 5n + 25q = 545 or .05n + .25q = 5.45 8 6/18/2015 n + q = 41 5n + 25q = 545 n + q = 41 5n + 25q = 545 n = 41 – q 5(41 – q) + 25q = 545 n = 41 – q 5(41 – q) + 25q = 545 205 – 5q + 25q = 545 205 + 20q = 545 q = 17 n + q = 41 5n + 25q = 545 n = 41 – q 5(41 – q) + 25q = 545 205 – 5q + 25q = 545 205 + 20q = 545 q = 17 n = 41 – 17 … n = 24 Another method for solving systems of equations is called linear combination. So you have 17 quarters and 24 nickels. The key to linear combination… This can also be called the addition/subtraction method. 9 6/18/2015 Linear combination (addition/subtraction) works well when you have terms that are the same or opposites. 5x + 3y = 7 5x + 7y = 3 4x + 2y = 20 4x – 2y = 4 1. Add or subtract the two equations to get one of the variables to cancel out. Steps for Linear Combination 0. If necessary, multiply one or both of the equations to get terms that are the same or opposites. (like 3x and -3x … you usually don’t have to do this step.) Solve with linear combination: 5x + 3y = 7 5x + 7y = 3 2. Solve the easy equation that’s left. 3. Substitute to get the other answer. Solve with linear combination: 5x + 3y = 7 – 5x + 7y = 3 Solve with linear combination: 5x + 3y = 7 – (5x + 7y = 3) -4y = 4 Subtract to cancel the x’s 10 6/18/2015 Solve with linear combination: 5x + 3y = 7 – (5x + 7y = 3) -4y = 4 Solve with linear combination: 5x + 3y = 7 – (5x + 7y = 3) -4y = 4 y = -1 Now solve for y. Solve with linear combination: 5x + 3y = 7 – 5x + 7y = 3 -4y = 4 y = -1 Substitute back in either original equation to find x. Solve with linear combination: 5x + 3y = 7 5x + 7y = 3 Solve with linear combination: 5x + 3y = 7 5x + 7y = 3 Solve with linear combination: 4x + 2y = 20 4x – 2y = 4 5x + 3(-1) = 7 5x – 3 = 7 5x = 10 … x=2 So … the answer is ( 2, -1) 11 6/18/2015 Solve with linear combination: 4x + 2y = 20 4x – 2y = 4 We could actually cancel out either the x’s or the y’s. It’s easiest to add the equations so the y’s cancel. Solve with linear combination: 4x + 2y = 20 4x – 2y = 4 x =3 4(3) + 2y = 20 12 + 2y = 20 2y = 8 … y = 4 Solve with linear combination: 4x + 2y = 20 + ( 4x – 2y = 4) 8x = 24 x =3 Solve with linear combination: 4x + 2y = 20 4x – 2y = 4 So … the answer is ( 3, 4 ) Another variation on the taco problem … Seven tacos and three burritos cost $17.34. Two tacos and three burritos cost $10.89. How much is a taco, and how much is a burrito? 7t + 3b = 17.34 2t + 3b = 10.89 12 6/18/2015 7t + 3b = 17.34 – (2t + 3b = 10.89) 5t = 6.45 t = 1.29 7(1.29) + 3b = 17.34 9.03 + 3b = 17.34 3b = 8.31 … b = 2.77 Tacos = $1.29 Burritos = $2.77 Remember Step #0 … 0. If necessary, multiply one or both of the equations to get terms that are the same or opposites. Here’s when you use it. 4x – 7y = 24 2x + 5y = -5 4x – 7y = 24 2x + 5y = -5 … 4x + 10y = -10 Nothing is exact the same or exact opposites. BUT … if you multiply the bottom by 2, the x’s will be the same. 4x – 7y = 24 4x + 10y = -10 4x – 7y = 24 – (4x + 10y = -10) -17y = 34 y = -2 13 6/18/2015 4x – 7y = 24 – (4x + 10y = -10) -17y = 34 y = -2 4x – 7(-2) = 24 4x + 14 = 24 4x = 10 … x = 2.5 So the answer is ( 2.5 , -2 ) 14