Solving Systems Using Substitution Objective: To solve systems algebraically Substitution Property of Equality • If a = b, then a can be used for b in any situation. • Ex: 1 = 2 2 So, if we’re adding 1 + 1 2 Then 2 + 1 = 3 we substituted the 2 for the 1 2 2 2 2 Solving for a Variable • Solving for x: • 1st … Eliminate the constant • Divide by the coefficient • Your answer is … 2x – 4 = 12 +4 +4 2x = 16 2 2 x=8 Solving for a Variable (Part II) • Solving for x: • 1st … Eliminate the constant • Divide by the coefficient • Your answer is … ax – b = c +b +b ax = c + b a a x=c+b a Systems of Equations • Two or more equations using the same two or more variables. Ex: 2x – y = 8 y=x–3 Notice that both equations have an x and a y Solving a System of Equations • Step 1: Find the variable that is already solved. • Step 2: If no variable is solved for, find out which one can be solve for easier. • Step 3: Insert the formula of the solved variable for the variable in the 2nd equation • Step 4: Solve for the 2nd Variable • Step 5: Use the answer to solve for the 1st variable • Step 6: Answer must come in an ordered pair (x, y) Example #1 Solve: 2x – y = 8 y=x–3 2x – (x – 3) = 8 2x – x + 3 = 8 x+3=8 -3 -3 x=5 • Step 1: Find the variable that is already solved • In this case, y is already solved for. y = x – 3. So use x – 3 for y in the top equation (Step 3) • Steps 4: Now solve for x Example #1 (Continued) x=5 y=x–3 y=5–3 y=2 (5,2) • Now plug in 5 for x in either equation. (Step 5) • Now set answer as an ordered pair (Step 6). Example #2 3x + y = 13 4x + 2y = 30 3x + y = 13 -3x -3x y = -3x + 13 4x + 2(-3x +13) = 30 4x – 6x + 26 = 30 -2x + 26 = 30 -26 -26 -2x = 4 -2 -2 x = -2 • Step #1: Find the variable that is already solve. • Not in this case. Go to Step 2. • To solve for x in equation #1, you need to subtract y and divide by 3. • To solve for y in equation #1, you need to subtract 3x and that’s it. • Once you find y, plug it in on the other equation. • Now solve for x • Now that we know x, plug it in either equation to find y. Example #2 (Continued) y = -3x + 13 y = -3(-2) + 13 y = 6 + 13 y = 19 (-2, 19) • We’ll use this equation, because it already gives us an answer for y. • Our solution