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Dr. Fowler CCM Solving Systems of Equations By Substitution – Easier 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: Plug 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. EXAMPLE 1 Solve by substitution: 2 x 7 y 12 x 2 y The second is solved for X. Substitute this into OTHER equation for X: 2 2 y 7 y 12 4 y 7 y 12 3 y 1 2 3 3 y 4 Substitute found y into other equation: x 2 y x 2 4 x 8 8, 4 The solution set found by the substitution method will be the same as the solution found by graphing. The solution set is the same; only the method is different. ALWAYS put answer in Alphabetical order. (x,y) 2) Solve the system using substitution x+y=5 y=3+x Step 1: Solve an equation for one variable. Step 2: Substitute Step 3: Solve the equation. The second equation is already solved for y! x+y=5 x + (3 + x) = 5 2x + 3 = 5 2x = 2 x=1 2) Solve the system using substitution x+y=5 y=3+x Step 4: Plug back in to find the other variable. Step 5: Check your solution. x+y=5 (1) + y = 5 y=4 (1, 4) (1) + (4) = 5 (4) = 3 + (1) The solution is (1, 4). What do you think the answer would be if you graphed the two equations? 3) Solve the system using substitution x=3–y x+y=7 Step 1: Solve an equation for one variable. Step 2: Substitute Step 3: Solve the equation. The first equation is already solved for x! x+y=7 (3 – y) + y = 7 3=7 The variables were eliminated!! This is a special case. Does 3 = 7? FALSE! 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. Step 2: Substitute Step 3: Solve the equation. The first equation is easiest to solved for y! y = -2x + 4 4x + 2y = 8 4x + 2(-2x + 4) = 8 4x – 4x + 8 = 8 8=8 This is also a special case. Does 8 = 8? TRUE! When the result is TRUE, the answer is INFINITELY MANY SOLUTIONS. Example 5) Solve the following system of equations using the substitution method. y = 3x – 4 and 6x – 2y = 4 The first equation is already solved for y. Substitute this into second equation. 6x – 2y = 4 6x – 2(3x – 4) = 4 6x – 6x + 8 = 4 8=4 (substitute) (use distributive property) (simplify the left side) Does 8=4? FALSE. Examples like this – the answer is NO SOLUTION Ø. If you graphed them, they would be PARALLEL LINES. EXAMPLE 6 Solve the system by the substitution method. 2 x 7 y 12 x 3 2y The second is solved for X. Substitute this into OTHER equation for X: 2 3 2 y 7 y 12 6 4 y 7 y 12 6 3 y 6 12 6 3 y 1 8 3 3 y 6 Substitute found y into other equation: x 3 2 6 x 3 12 x 15 15, 6 Example #7: y = 4x 3x + y = -21 Step 1: Solve one equation for one variable. y = 4x (This equation is already solved for y.) Step 2: Substitute the expression from step one into the other equation. 3x + y = -21 3x + 4x = -21 Step 3: Simplify and solve the equation. 7x = -21 x = -3 y = 4x 3x + y = -21 Step 4: We found x = -3. Now, substitute this into either original equation to find y: y = 4x (easiest) y = 4(-3) y = -12 Solution to the system is (-3, -12). Excellent Job !!! Well Done Stop Notes Do Worksheet