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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
• 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
• 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

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.)