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Solving Systems Using Substitution

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Title: Solving Systems with Substitution Author: Mike Meyer Last modified by: Mike Meyer Created Date: 7/17/2007 3:07:53 PM Document presentation format – PowerPoint PPT presentation

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Title: Solving Systems Using Substitution


1
7.2
  • Solving Systems Using Substitution

2
7.2 Solving Syst. By Subst.
  • Goal / I can
  • Solve systems using substitution

3
Solving Systems of Equations
  • You can solve a system of equations using
    different methods. The idea is to determine which
    method is easiest for that particular problem.
  • These notes show how to solve the system
    algebraically using SUBSTITUTION.

4
7.2 Solving Syst. By Subst.
  • Earlier this year we solved the following
  • y 2x 1 when x 4
  • We substituted x with 4 to make
  • y 2(4) 1
  • y 9

5
7.2 Solving Syst. By Subst.
  • The same idea can happen with linear systems.
  • Example
  • y 2x 2 y -3x 4
  • Since it says y , I can substitute.

6
7.2 Solving Syst. By Subst.
  • y 2x 2
  • -3x 4 2x 2 now I have 1 variable
  • -2x -2x
  • -5x 4 2
  • - 4 - 4
  • -5x -2
  • x .4 now substitute x to
    get y

7
7.2 Solving Syst. By Subst.
  • y 2(.4) 2
  • y .8 2
  • y 2.8
  • So my solution is (.4, 2.8) This is the other
    way to solve systems that I didnt mention
    yesterday.
  • Check you answer
  • 2.8 2(.4) 2
  • 2.8 2.8

8
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.
9
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
Step 3 Solve the equation.
10
1) 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?
11
Which answer checks correctly?
3x y 4 x 4y - 17
  1. (2, 2)
  2. (5, 3)
  3. (3, 5)
  4. (3, -5)

12
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
13
2) Solve the system using substitution
  • 3y x 7
  • 4x 2y 0

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

Step 5 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).
15
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

16
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.
17
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!
Step 3 Solve the equation.
When the result is TRUE, the answer is INFINITELY
MANY SOLUTIONS.
18
What does it mean if the result is TRUE?
  1. The lines intersect
  2. The lines are parallel
  3. The lines are coinciding
  4. The lines reciprocate
  5. I can spell my name
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