Dr. Fowler CCM

Solving Systems of EquationsBy Substitution – Easier

Solving a system of equations by substitution

Step 1: Solve an equation for one variable.

Step 2: Substitute

Step 3: Solve the equation.

Step 4: Plug back in to find the other variable.

Step 5: Check your solution.

Pick the easier equation. The goal

is to get y= ; x= ; a= ; etc.

Put the equation solved in Step 1

into the other equation.

Get the variable by itself.

Substitute the value of the variable

into the equation.

Substitute your ordered pair into

BOTH equations.

EXAMPLE 12 7 12

2

x y

x y

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)

Solve by substitution:

2 72 12yy 4 7 12y y

3 1

3 3

2y

4y

2x y

8x

42x

8, 4

Substitute found yinto other equation:

The second is solved for X. Substitutethis into OTHER equation for X:

2) Solve the system using substitution

x + y = 5

y = 3 + x

Step 1: Solve an equation for one variable.

Step 2: Substitute

The second equation is

already solved for y!

x + y = 5x + (3 + x) = 5

Step 3: Solve the equation.

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.

x + y = 5

(1) + y = 5

y = 4

Step 5: Check your solution.

(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 = 7Step 1: Solve an equation

for one variable.

Step 2: Substitute

The first equation is

already solved for x!

x + y = 7

(3 – y) + y = 7

Step 3: Solve the equation.

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 = 8Step 1: Solve an equation

for one variable.

Step 2: Substitute

The first equation is

easiest to solved for y!

y = -2x + 4

4x + 2y = 8

4x + 2(-2x + 4) = 8

Step 3: Solve the equation.

4x – 4x + 8 = 8

8 = 8This 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 (substitute)

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

8 = 4 (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

2 7 12

3 2

x y

x y

2 73 2 12yy 6 4 7 12y y

Solve the system by the substitution method.

6 3 126 6y 3 1

3 3

8y

6y

63 2x 3 12x 15x

15, 6

Substitute found yinto other equation:

The second is solved for X. Substitutethis into OTHER equation for X:

Example #7: y = 4x3x + 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 = 4x3x + 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

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