Unit 8 – Systems of Equations. System of Equation - more than one equation that is solved by 1) graphing or 2) substitution Graphing is easier, but the answers are hard to find Substitution is harder, but the answers are exact

3 types of answers A system has more than one equation for a problem. The solutions are either 1) One solution, an ordered pair (x, y) 2) no solutions, parallel line 3) infinite solutions, same line

LT 8-1 I can determine if an ordered pair is a solution of a system of equations 4x – y = 10 2x – y = 6 Is (2, -2) a solution of the system? 4(2) – (-2) = 10 2(2) – (-2)= 6 8 + 2 = 10 True 4 + 2 = 6 True Yes it is

Is (3, 6) a solution of the system? 3x -2y = 6 x = 3 3(3) -2(6) = 6 x = 3 9 -12 ≠ 6 False No cannot be a solution

Y = 1 4 x – y + 3x = 33 Is (12, 3) a solution of the system? 3 = 1 4 (12) – (3) + 3(12) = 33 True True It is a solution

LT 8.3 I can solve a system of equations by substitution https://tinyurl.com/vnerdsubst

Substitution Examples – just watch the process y = -2x – 3 y = 2x + 5 -2x – 3 = 2x + 5 - 3 = 4x + 5 -8 = 4x so x = -2 y = 2(-2) + 5 = 1 Solution is ( -2, 1), check in the other equation 1 = -2(-2) – 3 is true

Substitution Step 1 - If both equations = the same value (usually y), set them equal to each other and solve Step 2 – If not, solve one or both equations for a single variable and substitute in the other

2. y = -x + 7 y = 3x - 5 -x + 7 = 3x – 5 7 = 4x – 5 12 = 4x 3 = x y = 3(3) – 5 = 4 Solution is ( 3, 4) check in the other equation 4 = -(3) + 7 is true

5x -4(-3x + 5) = -3 Substitute for y 5x – 4y = -3 y = -3x + 5 5x -4(-3x + 5) = -3 Substitute for y 5x + 12x -20 = -3 17x – 20 = -3 17x = 17 x = 1 Find y y = -3(1) + 5 y = 2 Solution is (1, 2), check in 5(1) – 4(2) = -3 True

y = 3x y = 3x + 4 3x = 3x + 4 0 = 4 not true No Solutions

2y = 6x + 2 y - 2x = 1 Find the equation with a single variable and solve y - 2x = 1 y = 2x + 1 sub in for y in the red equation 2(2x + 1) = 6x + 2 4x + 2 = 6x + 2 2 = 2x + 2 0 = 2x x = 0 y = 2(0)+ 1 y = 1 (0,1)

LT 8.4 I can write and solve a system of equations. 3 steps Define two variables Write two equations Solve the system

Example

Step 1 a = adult c = children Step 2 a + c = 7 c + 1 = a Step 3 (c+ 1) + c = 7 2c + 1 = 7 2c = 6 c = 3 3 children and 1 adult