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Lesson 5'1, page 252 Systems of Linear Equations in Two Variables

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What if the equations are not 'easy' to graph? We can also ALGEBRA to solve systems by Substituting or the process of Elimination. ... – PowerPoint PPT presentation

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Title: Lesson 5'1, page 252 Systems of Linear Equations in Two Variables


1
Lesson 5.1, page 252Systems of Linear Equations
in Two Variables
  • Objective To solve systems of equations using
    substitution or elimination.

2
Definitions
  • System 2 or more equations together
  • Solution of system any ordered pair that makes
    all equations true
  • Possible Solutions one point, all points, no
    points

3
See Example 2, page 253.
  • Are the given ordered pairs solutions of the
    given systems?
  • x - y 17 and x y -1 (8,-9)
  • 3x 5y -12 and x - y 1 (-1,2)

4
Real World ConnectionRead Say Cheese! Pg. 252
  • Do you think graphing is the only way to solve a
    system?
  • What if the equations are not easy to graph?
  • We can also ALGEBRA to solve systems by
    Substituting or the process of Elimination.

5
Steps for using SUBSTITUTION
  • Solve one equation for one variable. (Hint
    Look for an equation already solved for one of
    the variables or for a variable with a
    coefficient of 1.)
  • Substitute into the other equation.
  • Solve this equation to find a value for a
    variable.
  • Substitute again to find the value of the other
    variable.
  • Check.

6
See Examples 3 4, page 254-5.
  • c) 2x y 6 d) x 6y - 2
  • y 5x x ¾y

7
What if one of the equations is not easily or not
already solved for a variable?
  • Use ELIMINATION!
  • Elimination method to use math operations to
    eliminate a variable

8
STEPS for ELIMINATION
  • 1) Look for a variable that has OPPOSITE
    coefficients. If found, ADD the two equations.
    Solve, substitute, solve.
  • 2) If no opposite coefficients, look for
    variables with the SAME coefficients. If found,
    SUBTRACT one equation from the other. Then,
    solve, substitute, solve.
  • Note Possible solutions are the same as before
  • One point in common
  • Infinitely many solutions
  • No solution

9
See Example 5, page 256.
  • e) x y -2 f) x 3y -5
  • x y 6 x 2y 0

10
What if neither variable can be eliminated by
simply adding or subtracting?
  • We may have to MULTIPLY before adding or
    subtracting!

11
Pencils down. Watch listen.
  • Solve using elimination.
  • 3x 5y 11
  • 2x 3y 7

12
Using Multiplication with Elimination
  • Write both equations in standard form.
  • Look for the easiest way to get one variable to
    have opposite coefficients. (Hint Think like
    youre finding an LCD).
  • Multiply EVERY term in the equation by the factor
    needed to get the opposites.
  • Follow the same steps for elimination with
    addition or subtraction.

13
See Example 6, page 257.
  • g) -2x 3y 1 h) 8x 4y 0
  • -4x y -3 4x 2y 2

14
What if theres fractions or decimals in the
equations?
  • Use the same process weve been using all
    semester to get rid of the fractions or decimals.
    (multiply by LCD or by powers of 10)

15
Fractional Coefficients
  • Solve using LCD method.

16
Special Cases
  • Dependent equations one equation is a multiple
    of the other Solution is INFINITELY MANY
    SOLUTIONS. (graph same line)
  • Note If completely solved, you get a true
    statement, like 0 0.
  • Inconsistent equations have NO SOLUTION (graph
    parallel lines)
  • Note If completely solved, you get a false
    statement, like 0 6.

17
See Examples 8 9, page 258-9.
  • j) 3x 2y 1 k) 8x 2y 2
  • -3x 2y -1 4x y 2

18
POSSIBLE SOLUTIONS for a System of Equations
  • Answer is a point, (x, y).
  • If variables cancel out and you get a true
    statement, the solution is all infinitely many
    solutions. (Graph would be the same line.)
  • If variables cancel out and you get a false
    statement, there is no solution. (Graph would be
    parallel lines.)
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