Solving Systems of Equations by Substitution (6-2) Objective: Solve systems of equations by using substitution. Solve real-world problems involving systems of equations by using substitution.

Solve by Substitution Another method of finding an exact solution of a system of equations is called substitution. Use the following steps when solving by substitution. 1.When necessary, solve at least one equation for one variable. 2.Substitute the resulting expression from Step 1 into the other equation to replace the variable. Then solve the equation. 3.Substitute the value from Step 2 into either equation, and solve for the other variable. Write the solution as an ordered pair (x, y).

Example 1 Use substitution to solve the system of equations. y = -4x x + y = 2 2x – 4x + 12 = 2 -2x + 12 = x = -10 x = 5 y = -4x + 12 y = -4(5) + 12 y = y = -8 Solution: (5, -8)

Check Your Progress Choose the best answer for the following. –Use substitution to solve the system of equations. y = 2x 3x + 4y = 11 A.(1, ½) B.(1, 2) C.(2, 1) D.(0, 0) 3x + 4(2x) = 11 3x + 8x = 11 11x = 11 x = 1 y = 2x y = 2(1) y = 2

Solve by Substitution If a variable is not isolated in one of the equations in a system, solve an equation for a variable first. Then you can use substitution to solve the system.

Example 2 Use substitution to solve the system of equations. x – 2y = -3 3x + 5y = 24 x – 2y = -3 +2y x = 2y – 3 3(2y – 3) + 5y = 24 6y – 9 + 5y = 24 11y – 9 = y = 33 y = 3 x = 2y – 3 x = 2(3) – 3 x = 6 – 3 x = 3 Solution: (3, 3)

Check Your Progress Choose the best answer for the following. Use substitution to solve the system of equations. 3x – y = x + 2y = 20 A.(-2, 6) B.(-3, 3) C.(2, 14) D.(-1, 8) 3x – y = x -y = -3x – 12 y = 3x x + 2(3x + 12) = 20 -4x + 6x + 24 = 20 2x + 24 = x = -4 x = -2 y = 3x + 12 y = 3(-2) + 12 y = y = 6

No Solution or Infinitely Many Solutions Generally, if you solve a system of equations and the result is a false statement such as 3 = -2, there is no solution. If the result is an identity, such as 3 = 3, then there are an infinite number of solutions.

Example 3 Use substitution to solve the system of equations. 2x + 2y = 8 x + y = -2 -y x = -y - 2 2(-y – 2) + 2y = 8 -2y – 4 + 2y = 8 -4 = 8 No Solution False

Check Your Progress Choose the best answer for the following. –Use substitution to solve the system of equations. 3x – 2y = 3 -6x + 4y = -6 A.One; (0, 0) B.No solution C.Infinitely many solutions D.Cannot be determined 3x – 2y = 3 +2y 3x = 2y + 3 x = 2 / 3 y ( 2 / 3 y + 1) + 4y = -6 -4y – 6 + 4y = = -6 True

Solve Real-World Problems You can use substitution to find the solution of a real-world problem involving a system of equations.

Example 4 A nature center charges $35.25 for a yearly membership and $6.25 for a single admission. Last week it sold a combined total of 50 yearly memberships and single admissions for $ How many memberships and how many single admissions were sold? x + y = x y = x + y = 50 -y x = -y (-y + 50) y = y y = y = y = y = 38 x = -y + 50 x = x = memberships and 38 single admissions were sold.

Check Your Progress Choose the best answer for the following. –Mikhail needs 10 milliliters of 25% HCl (hydorchloric acid) solution for a chemistry experiment. There is a bottle of 10% HCl solution and a bottle of 40% HCl solution in the lab. How much of each solution should he use to obtain the required amount of 25% HCl solution? A.0 mL of 10% solution, 10 mL of 40% solution B.6 mL of 10% solution, 4 mL of 40% solution C.5 mL of 10% solution, 5 mL of 40% solution D.3 mL of 10% solution, 7 mL of 40% solution x + y = 10 -y x = -y (-0.10y + 1) = 25(0.40y) -10y = 10y +10y 100 = 20y 5 = y x = -y + 10 x = x = 5