Solving Systems of Equations Using Substitution Lesson 6-2

4x – y = 75 (Since x = ______., ________ can be substituted The solution of a system of equation scan be found using one of 3 methods: ________________________ , _________________________, or by _________________________ (The purpose of the substitution method is that once you find one of the values (either x or y), you can then substitute it into either of the original equations to find the other value.) graphing substitution elimination Ex. A x = 4y 4x – y = 75 (Since x = ______., ________ can be substituted everywhere there is an “x” in the second equation.) 4y 4y NOW use x = 4y to find the value of x. (Substitute the answer you just got for “4”) x = 4y ___ ____________ Substitute _______________ Simplify 4x – y = 75 Second equation ____________ Substitute “4y” for “x” _____________ Simplify _____________ Combine like terms _____________ Divide both sides 4(4y) – y = 75 16y – y = 75 15y = 75 x = 4(5) 15y /15 = 75/15 x = 20 y= 5 The solution is ___________ (Don’t forget to write the solution, which is an ordered pair.) (5, 20)

________________ Second equation Ex. B Solve y = 2x 3x + 4y = 11     ________________ Second equation ________________ Substitute “2x” for “y” _______________ Simplify _______________ Combine like terms _________________________ Divide both sides ____________________ Simplify 3x + 4y = 11 Use the first equation to find the value of x. _______________ 1st equation _______________ Substitute _______________ Simplify 3x + 4(2x) = 11 3x + 8x = 11 y = 2x 11x = 11 y = 2(1) 11x/11= 11/11 y = 2 x = 1 The solution is ___________ (1, 2)

PRACTICE 1 x = 2y _____________________ ________________ 4x + 2y = 15 ____________________ ________________ ____________________ ________________   ____________________ ____________________ _____________________ The solution is ___________

PRACTICE 2 y = 3x - 8 _____________________ ________________ y = 4 – x ____________________ ________________ ____________________ ________________   ____________________ ____________________ _____________________ The solution is ___________

PRACTICE: 3 2x + 7y = 3 _____________________ ________________ x = 1 – 4y ____________________ ________________ ____________________ ________________   ____________________ ____________________ _____________________ The solution is ___________

-2x – 3y = 14 4x + y = 12 4x + y = 12 -2x – 3(12 – 4x) = 14 4x – 4x + y = 12 – 4x 4(5) + y = 12 -2x – 36 + 12x = 14 y = 12 – 4x 20+ y = 12 10x – 36 = 14 20+ y – 20 = 12 – 20 10x – 36+36 = 14 + 36 y = -8 10x = 50 10x/10 = 50/10 (5, -8) x = 5

You can check your answer using the graphing calculator OR by substituting the ordered pair into each of the equations to see if it is a solution for each equation.

2x + 2y = 8 2(2 – y) + 2y = 8 x + y = 2 4 – 2y + 2y = 8 x + y - y = 2 - y 4 = 8 x = 2 – y 4 = 8 false no solutions parallel lines

-6x +4y = -6 3x – 2y = 3 3x – 2y + 2y = 3 + 2y -2(3+ 2y) +4y = -6 3x = 3 + 2y (3x)/3 = (3 + 2y)/3 -6 – 4y +4y = -6 x = (3 + 2y)/3 -6 = -6 The statement ___________ is _________. There are ______ _______________________. The graphs of the lines are ________________________ -6 = -6 true infinitely many the same.

The school bookstore sells T-shirts for $8 and sweatshirts for $12 The school bookstore sells T-shirts for $8 and sweatshirts for $12. Last month, the store sold 37 T-shirts and sweatshirts for a total of $376. How many T-shirts were sold? x = number of T-Shirts sold Y = number of sweatshirts sold x + y = 37 8x + 12y = 376 x + y = 37 y = 37 - x 8x + 12 y = 376 8x + 12 (37 – x) = 376 8x + 444 – 12x = 376 -4x + 444 = 376 -4x + 444 – 444 = 376 – 444 -4x = -68 x = 17 Equation based on the number of items sold T-shirts were sold. (so 20 sweatshirts were sold) Equation based on the cost of items sold Check: 17(8 ) + 12(20) = 136 + 240 = 376

Watch the following videos for additional information on solving systems of equations using substitution. Solving systems using substitution Using Systems of Equations to Solve Real World Problems