Warm Up x = y – 3 9 – 3x 12 9 Simplify each expression. Solve each equation for x. 1. y = x + 3 2. y = 3x – 4 Simplify each expression. Evaluate each expression for the given value of x. 5. x + 8 for x = 6 6. 3(x – 7) for x = 10 x = y – 3 4. 12 – 3(x + 1) 3. 2(x – 5) 9 – 3x 2x – 10 2 3 12 9

Lesson 6.2 Solving Systems by Substitution

Objective Standards California 9.0 Students solve a system of two linear equations in two variables algebraically and are able to interpret the answer graphically. Student are able to solve a system of two linear inequalities in two variables and to sketch the solution sets.

Solving a system of equations by substitution Step 1: Solve an equation for one variable. Pick the easier equation. The goal is to get y= ; x= ; a= ; etc. Step 2: Substitute Put the equation solved in Step 1 into the other equation. Step 3: Solve the equation. Get the variable by itself. Step 4: substitute back in to find the other variable. Substitute the value of the variable into the equation. Step 5: Check your solution. Substitute your ordered pair into BOTH equations.

1) Solve the system using substitution x + y = 5 y = 3 + x Step 1: Solve an equation for one variable. The second equation is already solved for y! Step 2: Substitute x + y = 5 x + ( ) = 5 2x + 3 = 5 2x = 2 x = 1 -3 -3 Step 3: Solve the equation. Now you now the x-value is 1. Now lets find the value for y

1) Finding the y-value x + y = 5 y = 3 + x x + y = 5 + y = 5 y = 4 Step 4: substitute back in to find the other variable. 1 (1, 4) (1) + (4) = 5 (4) = 3 + (1) Step 5: Check your solution. The solution is (1, 4). What do you think the answer would be if you graphed the two equations?

Which answer checks correctly? 3x – y = 4 x = 4y - 17 (2, 2) (5, 3) (3, 5) (3, -5)

2) Solve the system using substitution 3y + x = 7 4x – 2y = 0 It is easiest to solve the first equation for x. 3y + x = 7 -3y -3y x = Step 1: Solve an equation for one variable. -3y + 7 Step 2: Substitute 4x – 2y = 0 4( ) – 2y = 0

2) Solve the system using substitution 3y + x = 7 4x – 2y = 0 4( -3x+ 7) – 2y = 0 -12y + 28 – 2y = 0 -14y + 28 = 0 -14y = -28 y = Step 3: Solve the equation. 2 4x – 2y = 0 4x – 2( ) = 0 4x – 4 = 0 4x = 4 x = 1 Step 4: Plug back in to find the other variable.

2) Solve the system using substitution 3y + x = 7 4x – 2y = 0 Step 5: Check your solution. (1, 2) 3(2) + (1) = 7 4(1) – 2(2) = 0 When is solving systems by substitution easier to do than graphing? When only one of the equations has a variable already isolated (like in example #1).

2x + 4y = 4 3x + 2y = 22 -4y + 4 -2y + 2 -2x + 4 -2y+ 22 If you solved the first equation for x, what would be substituted into the bottom equation. 2x + 4y = 4 3x + 2y = 22 -4y + 4 -2y + 2 -2x + 4 -2y+ 22

3) Solve the system using substitution x = x + y = 7 Step 1: Solve an equation for one variable. The first equation is already solved for x! Step 2: Substitute x + y = 7 ( ) + y = 7 3 = 7 The variables were eliminated!! This is a special case. Does 3 = 7? FALSE! Step 3: Solve the equation. When the result is FALSE, the answer is NO SOLUTIONS.

4) Solve the system using substitution 2x + y = 4 4x + 2y = 8 Step 1: Solve an equation for one variable. The first equation is easiest to solved for y! y = -2x + 4 4x + 2y = 8 4x + 2 ( ) = 8 Step 2: Substitute 4x – 4x + 8 = 8 8 = 8 This is also a special case. Does 8 = 8? TRUE! Step 3: Solve the equation. When the result is TRUE, the answer is INFINITELY MANY SOLUTIONS.

What does it mean if the result is “TRUE”? The lines intersect The lines are parallel The lines are coinciding

Application The sum of Maria’s sister and brother ages is 18. Her brother is four years older than her sisters. Find the ages of Maria’s sister and her brother. x + y = 18 Let x = her brother’s age Let y = her sister’s age x = 4 + y 4 + y + y = 18 X = 4 + 7 X = 11 Brother’s age = 11 Sister’s age = 7 4 + 2y = 18 -4 = -4 2y= 14 Y = 7

Lesson Quiz: Part I Solve each system by substitution. y = 2x (–2, –4) 1. 2. 3. y = 2x (–2, –4) x = 6y – 11 (1, 2) 3x – 2y = –1 –3x + y = –1 x – y = 4

8 hours; $480; plumber A: plumber A is cheaper for less than 8 hours. Lesson Quiz: Part II 4. Plumber A charges $60 an hour. Plumber B charges $40 to visit your home plus $55 for each hour. For how many hours will the total cost for each plumber be the same? How much will that cost be? If a customer thinks they will need a plumber for 5 hours, which plumber should the customer hire? Explain. 8 hours; $480; plumber A: plumber A is cheaper for less than 8 hours.