Goal 4 Objective 4.03 Using systems of linear equations or inequalities in two variables to model and solve problems.

1
Goal 4 Objective 4.03Using systems of linear
equations or inequalities in two variables to
model and solve problems.
2
Graphing Systems of Equations

• Graph of a System
• Intersecting lines- intersect at one point
• One solution
• Same Line- always are on top of each other, slope
  is equal. Infinitely many solutions

• Parallel lines- are opposite from each other and
  will never! intersect.
• No solution

3

• Example 1-Number of solutions
• Y-x5
• Yx-3
• Since the graph of y-x5 and yx-3 are
  intersecting lines there is one solution.
• Example 2-Solve a System of Equations
• Y-x8
• y4x-7
• the graph appears to intersect at the point with
  coordinates (3,5). Check the estimate by
  replacing x with 3 and 5 with y in both
  equations.

4
Substitution

• Example 1-Solve using Substitution
• Y3x
• X2y-21
• Since y 3x, substitute 3x for y in the second
  equation.
• X2y-21 2nd equation
• X2(3x)-21 y3x
• x6x-21 simplify
• 7x-21 combine like terms
• 7x/7-21/7 divide each side by 7
• X-3 Use y3x to find
  the value of y.
• y3x y-9
• Y3(-3) The solution is ( -3,-9)
  

5
Elimination Using Addition and Subtraction

• Example 1-Elimination using Addition
• 3x-5y-16
• 2x5y31
• Since the coefficients of the y terms,-5 and 5,
  are additive inverse, you can eliminate the y
  terms by adding the equations.
• 3x-5y-16
• 2x5y31 Notice the y variable is
  eliminated.
• 5x 15
• 5 5 divide each side by 5
• x3 simplify

6

• Example 1-Continued
• Now substitute 3 for x in either equation to find
  the value of y.
• 3x-5y-16 first equation
• 3(3)-5y-16 replace x with 3
• 9-5y-16 simplify
• -9 -9 subtract 9 from each side
• -5y-25 simplify
• -5 -5 divide each side by -5
• Y5
• The solution is (3, 5)

7

• Example 2-Elimination Using Subtraction
• 5s2t6
• 9s2t22
• Since the coefficients of the t terms, 2 and 2,
  are the same you can eliminate them by using
  subtraction.
• 5s2t6
• (-)9s2t22 Notice that the variable t is
  eliminated.
• -4s -16
• -4 -4 Divide each side by -4
• S4 Simplify

8

• Example 2- Continued
• Now substitute 4 for s in either equation to find
  the value of y.
• 5s2t6
• 5(4)2t6 Replace s with 4
• 20 2t6 Simplify
• -20 -20 Subtract 20 from each side
• 2t-14 Simplify
• 2 2 Divide each side by 2
• t-7 The solution is (4, -7)

9
Elimination Using Multiplication

• Example 1-Multiply one Equation to Eliminate
• 3x4y6
• 5x2y-4
• Multiply the 2nd equation by -2 so the
  coefficients of the y terms are additive inverse.
  Then add the equations.
• 3x4y6 3x4y6
• 5x2y-4 ()-10x-4y8
  
• Mult by 2
  -7x 14 Add the equations
• -7
  -7 Divide each side by -7
• 
  x-2

10

• Example 1- Continued
• Now substitute -2 for x in either equation.
• 3x4y6
• 3(-2)4y6 x-2
• -64y6 Simplify
• 6 6 add 6 to both sides
• 4y12 Simplify
• 4 4 Divide by 4 on each side
• y3 The solution is (-2, 3)

11
The end