15.3 Solving Systems of Linear Equations by Elimination

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15.3 Solving Systems of Linear Equations by
Elimination
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• A system of linear equations is composed of 2
  equations using 2 variables x and y. The solution
  to a system is the ordered pair (x,y) that makes
  both equations true.

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For example (1,3) is a solution to the system2x
3y 115x - y 2because the x-value of 1 and
y-value of 3 makes both equations true.
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To solve the system

• sketch the graph of each equation on the same
  axes.

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Since the graphs intersect at the point (4, 2)
that is the solution to the system.
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When graphing a system 3 things can occur

• (1) the lines intersect in only 1 point like the
  last problem. In this case the system has only
  one solution which is the ordered pair

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(2) the lines are parallel and do not intersect
at all.In this case the system has no solutions
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(3) the lines coincide (they are the same
points). In this case the system has an infinite
number of solutions.
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There are 3 methods you can use to solve a system.

• Graphing- sketch the graph of each equation
• Substitution
• Elimination (addition)

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In this section we will look at solving a system
using the elimination method. This method works
best when the equations of the system are written
in standard form and have opposite coefficients
for either x or y.
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Elimination (addition)Since each equation is in
standard form, add the like terms of the
equations causing y to drop out .

• x y 5
• 3x - y 3

The solution is (2, 3)
4x
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Solving this equation for x gives x 2. What is
y? y 3
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Use elimination to solve each system.
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If the equations are in standard form and no
coefficients are opposites, then multiply one or
both equations by numbers to obtain a system that
does. This can be done many different ways.
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Example Solve the system by eliminating the y
variable.

• Since the y coefficients are not opposites, we
  must find a value to multiply one equation or
  both so that the coefficients are opposites.
• Multiply the first equation by -2
• Add the equations
• Since x 5, place that into either equation and
  find y.
• 2(5) y 3
• y -7
• Solution is (5, -7)

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Solve the system by elimination.
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Solve each system by elimination.