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**Solving Systems of Equations By Substitution – Easier**

Dr. Fowler CCM Solving Systems of Equations By Substitution – Easier

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**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.

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**EXAMPLE 1 Solve by substitution:**

The second is solved for X. Substitute this into OTHER equation for X: Substitute found y into other equation: 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)

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**2) 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 + (3 + x) = 5 2x + 3 = 5 2x = 2 x = 1 Step 3: Solve the equation.

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**2) Solve the system using substitution**

x + y = 5 y = 3 + x x + y = 5 (1) + y = 5 y = 4 Step 4: Plug back in to find the other variable. (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?

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**3) Solve the system using substitution**

x = 3 – y 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 (3 – y) + 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.

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**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(-2x + 4) = 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.

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**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 (substitute) 6x – 6x + 8 = 4 (use distributive property) 8 = (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.

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**EXAMPLE 6 Solve the system by the substitution method.**

The second is solved for X. Substitute this into OTHER equation for X: Substitute found y into other equation:

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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

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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).

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**Excellent Job !!! Well Done**

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