7.2: Solving Systems of Equations using Substitution - PowerPoint PPT Presentation

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7.2: Solving Systems of Equations using Substitution

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Title: 7.2: Solving Systems of Equations using Substitution


1
7.2 Solving Systems of Equations using
Substitution
2
Solving Systems of Equations using Substitution
Steps 1. Solve one equation for one
variable (y x a) 2. Substitute the
expression from step one into the other
equation, and SOLVE. 3. Substitute back into
the equation we solved for in Step 1, and
SOLVE 4. Check the solution in both equations of
the system.
3
Example 1
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
7x -21 x -3
4

y 4x 3x y -21
Step 4 Substitute back into either original
equation to find the value of the
other variable. y 4x
4 (-3) y -12
Solution to the system is (-3, -12).
5

y 4x 3x y -21
Step 5 Check the solution in both equations.
Solution to the system is (-3,-12).
3x y -21 3(-3) (-12) -21 -9
(-12) -21 -21 -21
y 4x -12 4(-3) -12 -12
6
Example 2
x y 10 5x y 2
Step 1 Solve one equation for one variable.
x y 10 y
-x 10
Step 2 Substitute the expression from step one
into the other equation. 5x - y 2
5x -(-x 10) 2
7

x y 10 5x y 2
Simplify!!
5x -(-x 10) 2 5x x -10 2 6x
-10 2 6x 12 x
2
8

x y 10 5x y 2
Step 4 Substitute back into the equation we
solved for in step 1 y -x 10 -(2)
10 y 8
Solution to the system is (2,8).
9

x y 10 5x y 2
Step 5 Check the solution in both equations.
Solution to the system is (2, 8).
5x y 2 5(2) - (8) 2 10 8 2 2 2
x y 10 2 8 10 10 10
10
Solving a system of equations by substitution
Pick the easier equation. The goal is to get y
x a etc.
  • Step 1 Solve an equation for one variable.

Step 2 Substitute
Put the equation solved in Step 1 into the other
equation.
Substitute the value of the variable into the
equation.
Step 3 Plug back in to find the other variable.
Step 4 Check your solution.
Substitute your ordered pair into BOTH equations.
11
1) 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
12
1) Solve the system using substitution
  • x y 5
  • y 3 x

x y 5 (1) y 5 y 4
Step 3 Plug back in to find the other variable.
(1, 4) (1) (4) 5 (4) 3 (1)
Step 4 Check your solution.
The solution is (1, 4). What do you think the
answer would be if you graphed the two equations?
13
Which answer checks correctly?
3x y 4 x 4y - 17
  1. (2, 2)
  2. (5, 3)
  3. (3, 5)
  4. (3, -5)

14
2) Solve the system using substitution
  • 3y x 7
  • 4x 2y 0

It is easiest to solve the first equation for
x. 3y x 7 -3y -3y x -3y 7
Step 1 Solve an equation for one variable.
Step 2 Substitute
4x 2y 0 4(-3y 7) 2y 0
15
2) Solve the system using substitution
  • 3y x 7
  • 4x 2y 0

-12y 28 2y 0 -14y 28 0 -14y -28 y 2
4x 2y 0 4x 2(2) 0 4x 4 0 4x 4 x 1
Step 3 Plug back in to find the other variable.
16
2) Solve the system using substitution
  • 3y x 7
  • 4x 2y 0

Step 4 Check your solution.
(1, 2) 3(2) (1) 7 4(1) 2(2) 0
When is solving systems by substitution easier to
do than graphing? When only one of the equations
has a variable already isolated (like in example
1).
17
If you solved the first equation for x, what
would be substituted into the bottom equation.
2x 4y 4 3x 2y 22
  1. -4y 4
  2. -2y 2
  3. -2x 4
  4. -2y 22

18
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!
When the result is FALSE, the answer is NO
SOLUTIONS.
19
3) 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!
When the result is TRUE, the answer is INFINITELY
MANY SOLUTIONS.
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