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Introduction
Two equations that are solved together are called
systems of equations. The solution to a system of
equations is the point or points that make both equations
true. Systems of equations can have one solution, no
solutions, or an infinite number of solutions. Finding the
solution to a system of equations is important to many
real-world applications.
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2.2.1: Proving Equivalencies
Key Concepts
• There are various methods to solving a system of
equations. Two methods include the substitution
method and the elimination method.
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2.2.1: Proving Equivalencies
Key Concepts, continued
Solving Systems of Equations by Substitution
• This method involves solving one of the equations for
one of the variables and substituting that into the
other equation.
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2.2.1: Proving Equivalencies
Key Concepts, continued
Substitution Method
1. Solve one of the equations for one of the variables in
terms of the other variable.
2. Substitute, or replace the resulting expression into
the other equation.
3. Solve the equation for the second variable.
4. Substitute the found value into either of the original
equations to find the value of the other variable.
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2.2.1: Proving Equivalencies
Key Concepts, continued
• Solutions to systems are written as an ordered pair,
(x, y). This is where the lines would cross if graphed.
• If the resulting solution is a true statement, such as 9
= 9, then the system has an infinite number of
solutions. This is where lines would coincide if
graphed.
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2.2.1: Proving Equivalencies
Key Concepts, continued
• If the result is an untrue statement, such as 4 = 9,
then the system has no solutions. This is where lines
would be parallel if graphed.
• Check your answer by substituting the x and y values
back into the original equations. If the answer is
correct, the equations will result in true statements.
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2.2.1: Proving Equivalencies
Key Concepts, continued
Solving Systems of Equations by Elimination Using
Addition or Subtraction
• This method involves adding or subtracting the
equations in the system so that one of the variables is
eliminated.
• Properties of equality allow us to combine the
equations by adding or subtracting the equations to
eliminate one of the variables.
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2.2.1: Proving Equivalencies
Key Concepts, continued
Elimination Method Using Addition or Subtraction
1. Add the two equations if the coefficients of one of the
variables are opposites of each other.
2. Subtract the two equations if the coefficients of one
of the variables are the same.
3. Solve the equation for the second variable.
4. Substitute the found value into either of the original
equations to find the value of the other variable.
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2.2.1: Proving Equivalencies
Key Concepts, continued
Solving Systems of Equations by Elimination Using
Multiplication
• This method is used when one set of variables are
neither opposites nor the same. Applying the
multiplication property of equality changes one or both
equations.
• Solving a system of equations algebraically will
always result in an exact answer.
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2.2.1: Proving Equivalencies
Key Concepts, continued
Elimination Method Using Multiplication
1. Multiply each term of the equation by the same
number. It may be necessary to multiply the second
equation by a different number in order to have one
set of variables that are opposites or the same.
2. Add or subtract the two equations to eliminate one of
the variables.
3. Solve the equation for the second variable.
4. Substitute the found value into either of the original
equations to find the value of the other variable.
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2.2.1: Proving Equivalencies
Common Errors/Misconceptions
• finding the value for only one of the variables of the
system
• forgetting to distribute negative signs when
substituting expressions for variables
• forgetting to multiply each term by the same number
when solving by elimination using multiplication
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2.2.1: Proving Equivalencies
Guided Practice
Example 2
Solve the following system by elimination.
ì2x - 3y = -11
í
î x + 3y = 11
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2.2.1: Proving Equivalencies
Guided Practice: Example 2, continued
1. Add the two equations if the coefficients of
one of the variables are opposites of each
other.
3y and –3y are opposites, so the equations can be
added. Add downward, combining like terms only.
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2.2.1: Proving Equivalencies
Guided Practice: Example 2, continued
2x - 3y = -11
x + 3y = 11
3x + 0 =
0
Simplify.
3x = 0
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2.2.1: Proving Equivalencies
Guided Practice: Example 2, continued
2. Solve the equation for the second
variable.
3x = 0
x=0
Divide both sides by 3.
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2.2.1: Proving Equivalencies
Guided Practice: Example 2, continued
3. Substitute the found value, x = 0, into
either of the original equations to find the
value of the other variable.
2x – 3y = –11
First equation of the system
2(0) – 3y = –11 Substitute 0 for x.
– 3y = –11
y=
11
3
Simplify.
Divide both sides by –3.
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2.2.1: Proving Equivalencies
Guided Practice: Example 2, continued
4. The solution to the system of equations
æ 11ö
is ç 0, ÷ .
è 3ø
æ 11ö
If graphed, the lines would cross at ç0, ÷.
è 3ø
✔
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2.2.1: Proving Equivalencies
Guided Practice: Example 2, continued
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2.2.1: Proving Equivalencies
Guided Practice
Example 3
Solve the following system by multiplication.
ì x - 3y = 5
í
î-2x + 6y = 4
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2.2.1: Proving Equivalencies
Guided Practice: Example 3, continued
1. Multiply each term of the equation by the
same number.
The variable x has a coefficient of 1 in the first
equation and a coefficient of –2 in the second
equation.
Multiply the first equation by 2.
x – 3y = 5
2(x – 3y = 5)
2x – 6y = 10
Original equation
Multiply the equation by 2.
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2.2.1: Proving Equivalencies
Guided Practice: Example 3, continued
2. Add or subtract the two equations to
eliminate one of the variables.
2x - 6y = 10
+(-2x + 6y = 4)
0 + 0 = 14
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2.2.1: Proving Equivalencies
Guided Practice: Example 3, continued
3. Simplify.
0 + 0 = 14
0 = 14
This is NOT a true statement.
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2.2.1: Proving Equivalencies
Guided Practice: Example 3, continued
ì x - 3y = 5
4. The system í
does not have a
î-2x + 6y = 4
solution. There are no points that will
make both equations true.
✔
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2.2.1: Proving Equivalencies
Guided Practice: Example 3, continued
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2.2.1: Proving Equivalencies