Do NOW
Simplify.
1.
2.
3.
4r + 3(r+5)
z – 2( Z - 2)
5(4p – 7) – 2p
• 1. 7r+15
• 2. -z + 4
• 3. 18p - 35
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The Substitution Method
Coordinate Algebra
Standard: A.REI.5
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The Substitution Method
Essential Question
• How do I prove that a system of two
equations in two variables can be solved by
multiplying and adding to produce a system
with the same solutions?
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Systems of Equations
Terms
System of Equations: two equations in two
variables.
Solution to a System: ordered pair that is a
solution to all equations in the system.
The answer to a system is the point of
intersection for the two lines!
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Other Solution Types:
• If the lines in the system intersect, that POINT
is the solution to the system.
• If the lines do not intersect (they are parallel)
then the answer is “No Solution”.
• If the lines are equivalent, which means they
graph the same line, then the answer is
“Infinitely Many Solutions”.
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The Substitution Method
Glossary Term
Substitution Method: method used to solve a
system of equations in which variables are
replaced with known values or algebraic
expressions.
Need either x= or y= to use this method.
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Ex. 1) Solve using substitution.
8 x  2 y  19

x  3
Since you know x = 3 from
looking at the second
equation…..
8(3) + 2y = 19
24 + 2y = 19
2y = 19 –24
2y = -5
y = -5/2
SUBSTITUTE 3 in for x in
the first equation.
(3, -5/2)
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Ex. 2) Solve using substitution.
Use this system.
Notice the second
equation is solved for y.
(so y is the same thing
as 2x+3)
Therefore, just replace y
with 2x+3 in the first
equation.
*remember to use ( ) each time
you substitute! It will make a
difference!!!
15 x  5 y  30

 y  2x  3
15x – 5(2x + 3) = 30
15x –10x –15 = 30
5x - 15
= 30
5x
= 45
x
=9
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X=9
Now you have to find y!
15 x  5 y  30

 y  2x  3
Choose either equation
to find y.
The solution to the
system is (9,21)
(the 2nd is easiest since its
solved for y!)
** On the graph, this is
where they would
intersect!!!
Y = 2(9) + 3
Y = 21
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The Substitution Method
Ex. 3) Solve using substitution.
Find an exact solution to a system of
equations by using the substitution method.
x+y=7
x + 2y = –1
Solve the first
equation for x.
x+y=7
x = –y + 7
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The Substitution Method
Key Skills
Find an exact solution to a system of
equations by using the substitution method.
x+y=7
x + 2y = –1
Substitute for x in the
second equation and
solve for y.
(the other equation
from the one you used
to find the value!)
x = –y + 7
x + 2y = –1
(–y + 7) + 2y = –1
7 + 2y – y = –1
7 + y = –1
y = –8
Copyright © by Holt, Rinehart and Winston. All Rights Reserved.
The Substitution Method
Key Skills
Find an exact solution to a system of
equations by using the substitution method.
x+y=7
x + 2y = –1
x = 15
y = –8
+y=7
SubstituteThe
for ysolution
in
is (15,x–8).
either of the original
x + (–8) = 7
equations and
x = 15
solve for x.
TOC
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Ex. 4) Solve by substitution.
Your turn.
You are lucky here.
2 x  5 y  14

You KNOW y is 5.
y  5
Plug 5 in for y in to
your first equation.
2x + 5(5) = 14
2x + 25 = 14
(-11/2, 5) is the answer.
2x = -11
x = -11/2
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Ex. 5) Solve.
• This time, its not so
easy to see the
value of one of the
variables.
• We pick either
equation and solve
for either variable.
• Which one would
make sense here to
use and for x or y?
3x  y  4

5 x  7 y  11
It makes since to use
the 1st equation and
solve for y… (it has
one as the coefficient!)
Y = -3x + 4
In other words, y is the
same thing as –3x + 4
Copyright © by Holt, Rinehart and Winston. All Rights Reserved.
3x  y  4

5 x  7 y  11
Y = -3x + 4
So use –3x + 4 for y in the
second equation. (the other
equation from the one you
chose to rewrite.)
5x – 7(-3x + 4) = 11
5x + 21x – 28 = 11
26x - 28 = 11
26x = 39
x = 39/26
x = 3/2
Now sub 3/2 in for
x in either
equation to find y.
3(3/2) + y = 4
9/2 + y = 4
(4 ½) + y = 4
y = -1/2
Solution (3/2, -1/2)
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Summarizer
How many solution types can you have when
solving a system of linear equations?
THREE
What are they?
One Solution (ordered pair), No Solution
(parallel lines), and Infinitely Many Solutions
(lines are the same).
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Homework
• Worksheet
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