Solving Systems of Equations

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Solving Systems of Equations

• Modeling real-world problems

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In business, the point at which income equals
expenses is called the break-even point.When
starting a business, people want to know the
point a which their income equals their expenses,
thats the point where they start to make a
profit. In the example above the values of y on
the blue line represent dollars made and the
value of y on the dotted red line represent
dollars spent.
Profit
Loss
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A system of equations is a set of two or more
equations that have variables in common.

• The common variables relate to similar
  quantities. You can think of an equation as a
  condition imposed on one or more variables, and a
  system as several conditions imposed
  simultaneously.
• Remember, when solving systems of equations, you
  are looking for a solution that makes each
  equation true.

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In earlier chapters, you learned to solve an
equation for a specified variable, graph the
equation, and find how pairs of lines represented
by linear equations in two variables are related.

• When solving a system of equations, you look for
  a solution that makes each equation true. There
  are several strategies you can use. To begin with
  we will be using tables and graphs. Lets look
  at the following example and work through the
  problem step-by-step to find a solution.

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Edna leaves the trailhead at dawn to hike 12
miles toward the lake, where her friend Maria is
camping. At the same time, Maria starts her hike
toward the trailhead. Edna is walking uphill so
she averages only 1.5 mi/hr, while Maria averages
2.5 mi/hr walking downhill. When and where will
they meet?
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Heres what we need to complete to solve this
example.

• Define variables for time (x) and for distance
  (y) from the trailhead.
• Write a system of two equations to model this
  situation.
• Solve this system by creating a table and finding
  values for the variables that make both equations
  true. Then locate this solution on a graph.
• Check your solution and explain its real-world
  meaning.

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Let x represent the time in hours. Both women
hike the same amount of time. Let y represent
the distance in miles from the trailhead. When
Edna and Maria meet they will both be the same
distance from the trailhead, although they will
have hiked different distances.
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The system of equations that models this
situation is grouped in a brace.Edna starts at
the trailhead so she increases her distance from
it as she hikes 1.5 mi/hr for x hours. Maria
starts 12 miles from the trailhead and reduces
her distance from it as she hikes 2.5 mi/hr for x
hours.
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Create a table from the equations. Fill in the
times and calculate each distance. The table
shows the x-value that gives equal y-values for
both equations. When x 3, both y-values are
4.5. So the solution is the ordered pair (3,
4.5). We say these values satisfy both
equations.
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Lets create the data on your graphing
calculator. Enter the equations in y on your
calculator and create a table.
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Do both tables model this situation?What do you
notice about y when x increases in each
equation?Why are the values different?
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On the graph this solution is the point where the
two lines intersect. You can use trace function
or calculate function on your calculator to
approximate the coordinates of the solution
point, though sometimes youll get an exact
answer as in our example here.
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Solving Systems of Equations

• Graphing Method

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A system of linear equations is a set of two or
more equations with the same variable. The
solution of a system in x and y is any ordered
pair (x, y) that satisfies each of the equations
in the system.The solution of a system of
equations is the intersection of the graphs of
the equations.
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If you can graph a straight line, you can solve
systems of equations graphically!The process is
very easy. Simply graph the two lines and look
for the point where they intersect
(cross).Remember using the graphing method many
times only approximates the solution, so
sometimes it can be unreliable.
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Solving Systems of Equations by Graphing.To
solve a system of equations graphically, graph
both equations and see where they intersect. The
intersection point is the solution.

• 4x 6y 12
• 4x 6y 12
• 4x 12 6y
• 6y 4x 12
• 6 6 6
• y 2/3x 2
• slope 2/3
• y-intercept -2

• 2x 2y 6
• 2y -2x 6
• 2 2 2
• y -x 3
• slope -1/1
• y-intercept 3

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Graph the equations.The slope-intercept method
of graphing was used in this example.The point
of intersection of the two lines (3, 0) is the
solution to the system of equations.This means
that (3, 0), when substituted into either
equation, will make them both true.
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Use a graph to solve the system of equations
below. Graph both equations on the same
coordinate plane.Graph x y 5 using the
intercepts (5, 0) and (0, 5)Graph y 2x 1
using the slope-intercept method.
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Locate the point where the lines intersect. From
the graph, the solution appears to (2, 3).Check
to be sure that (2, 3) is the solution,
substitute 2 for x and 3 for y into each
equation.x y 5 y 2x 12 3 5 3
2(2) - 1
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Use a graph to solve each system of equations.
If the system has no solution, write none.
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Summary of Solutions of Systems of Linear
Equations
The lines intersect so there is one solution.
y 2x 7 Y 2x 3
The lines are parallel so there are no solutions.
x 2y 7 x y 4
The lines are the same so there are infinitely
many solutions.
-3x 5 y 2y 6x 10
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NoteSome systems of equations may be very
difficult to solve using the graphing method.
The exact solution would be hard to determine
from a graph because the coordinates are not
integers. Solving a system algebraically is
better than graphing when you need an accurate
solution.Take for example the system of
equations3x 2y 12x y 3x 3/8 and y
15/4
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Solving Systems of Equations Algebraically

• Substitution Method

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The substitution method is used to eliminate one
of the variables by replacement when solving a
system of equations.Think of it as grabbing
what one variable equals from one equation and
plugging it into the other equation.
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Solve this system of equations using the
substitution method.

• Step 1
• 3y 2x 11
• Y 2x 9
• Solve one of the equations for either x or y.
• In this example it is easier to solve the second
  equation for y, since it only involves one
  step.
• Y 9 2x

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Step 2Replace the y value in the first
equation by what y now equals (y 9 2x).
Grab the y value and plug it into the other
equation.3y 2x 113(9 2x) 2x 11
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Step 3Solve this new equation for x.3(9
2x) 2x 1127 6x 2x 1127 6x 2x
1127 8x 11-8x -16x 2
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Step 4Now that we know the x value (x 2),
we place it into either of the ORIGINAL equations
in order to solve for y. Pick the easier one
to work with!Y 2x 9y 2(2) 9y 4
9y 5
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Step 5Check substitute x 2 and y 5 into
BOTH ORIGINAL equations. If these answers are
correct BOTH equations will be true!3y 2x
113(5) 2(2) 1115 4 11 11 11 True?Y
2x 95 2(2) 95 4 99 9 True?
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The Substitution Method

• Step 1 Solve one equation for x (or y).
• Step 2 Substitute the expression from Step 1
  into the other equation.
• Step 3 Solve for y (or x).
• Step 4 Take the value of y (or x) found in Step
  3 and substitute it into one of the original
  equations. Then solve for the other variable.
• Step 5 The ordered pair of values from Steps 3
  and 4 is the solution. If the system has no
  solution, a contradictory statement will result
  in either Step 3 or 4.

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Use the substitution method to solve each system
of equations. Check your answers.
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Solving systems of equations in real-world
problems.

• April sold 75 tickets to a school play and
  collected a total of 495. If the adult tickets
  cost 8 each and child tickets cost 5 each, how
  many adult tickets and how many child tickets did
  she sell?
• Solution Let a represent the adult tickets and
  c represent the child tickets.
• Individual tickets sold equaled 75, so a c 75
• All total April sold 495 in tickets, since adult
  tickets are 8 and child tickets are 5, so 8a
  5c 495.

System of Equations a c 75 8a 5c 495
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Solutiona c 758a 5c 495a 75 c8(75
c) 5c 495600 8c 5c 495600 - 3c
495105 3c35 ca c 75a 35 75a
40There were 40 adult tickets and 35 child
tickets sold. 40 35 758(40) 5(35)
495320 175 495
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Write a system of equations and solve.At a
baseball game, Jose bought five hot dogs and
three sodas for 17. At the same time, Allison
bought two hot dogs and four sodas for 11. Find
the cost of one hot dog and one soda.
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Solving Systems of Equations

• Elimination method

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You can use the Addition and Subtraction
Properties of Equality to solve a system by the
elimination method. You can add or subtract
equations to eliminate (getting rid of) a
variable.

• Step 1
• 5x 6y -32
• 3x 6y 48
• Eliminate y because the sum of the coefficients
  of y is zero
• 5x 6y -32
• 3x 6y 48
• 8x 0 16
• x 2

If you add the two equations together, the 6y
and -6y cancel each other out because of the
Property of Additive Inverse
Addition Property of Equality
Solve for x
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Step 2Solve for the eliminated variable y using
either of the original equations.3x 6y
483(2) 6y 486 6y 486y 42y 7
Choose the 2nd equation
Remember x 2
Substitute 2 for x
Simplify. Then solve for y.
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Since x 2 and y 7, the solution is (2, 7)

• Check
• 5x 6y -32 3x 6y 48
• 5(2) 6(7) -32 3(2) 6(7) 48
• 10 42 -32 6 42 48
• -32 -32 48 48
• Remember, the order pair (2, 7) must make both
  equations true.

True
True
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Suppose your community center sells a total of
292 tickets for a basketball game. An adult
ticket cost 3. A student ticket cost 1. The
sponsors collected 470 in ticket sales. Write
and solve a system to find the number of each
type of ticket sold.
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Let a number of adult ticketsLet s number of
student ticketstotal number of ticket total
number of salesa s 292 3a
1s 470Solve by elimination (get rid of s)
because the difference of the coefficients of s
is zero.a s 2923a s 470-2a 0
-178a 89
This is the number of adult tickets sold.
That means you must subtract the two equations
so, -3a a -470 is what you must subtract.
Next Step
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Solve for the eliminated variable using either of
the original equations.a s 29289 s
292s 203There were 89 adult tickets sold and
203 student tickets sold.Is the solution
reasonable? The total number of tickets is 89
203 292. The total sales is 3(89) 1(203)
470. The solution is correct.
This is the number of student tickets sold.
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If you have noticed in the last few examples that
to eliminate a variable its coefficients must
have a sum or difference of zero.

• Sometime you may need to multiply one or both of
  the equations by a nonzero number first so that
  you can then add or subtract the equations to
  eliminate one of the variables.
• 2x 5y 17 7x 2y 10 2x 5y -22
• 6x 5y -19 -7x y -16 10x 3y 22
• If you notice the systems of equations above, two
  of them have something in common. The third
  doesnt.

We can add these two equations together to
eliminate the y variable.
We can add these two equations together to
eliminate the x variable.
What are we going to do with these equations,
cant eliminate a variable the way they are
written?
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Multiplying One Equation
Be careful when you subtract. All the signs in
the equation that is being subtracted
change. -10x 3y -22

• Solve by Elimination
• 2x 5y -22
• 10x 3y 22
• Step 1
• 2x 5y -22 5(2x 5y -22) 10x 25y -110
• 10x 3y 22 10x 3y 22 -(10x 3y 22)
• 
  0 22y -132
• 
  y -6

To prepare for eliminating x, multiply the first
equation by 5.
Subtract the equations to eliminate x.
Start with the given system.
Ask Is one coefficient a factor of the other
coefficient for the same variable?
NEXT
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Step 2Solve for the eliminated variable using
either of the original equations. 2x 5y
-22 Choose the first equation.2x 5(-6) -22
Substitute -6 for y. 2x 30 -22 Solve for
x. 2x 8 x 4The
solution is (4, -6).
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Solve by elimination.-2x 5y -327x 5y
173x 10y -254x 40y 202x 3y 612x
y -7
Ask Is one coefficient a factor of the other
coefficient for the same variable?
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Multiplying Both Equations

• To eliminate a variable, you may need to multiply
  both equations in a system by a nonzero number.
  Multiply each equation by values such that when
  you write equivalent equations, you can then add
  or subtract to eliminate a variable.
• 4x 2y 14
• 7x 3y -8

In these two equations you cannot use graphing or
substitution very easily. However ever if we
multiply the first equation by 3 and the second
by 2, we can eliminate the y variable.
Find the least common multiple LCM of the
coefficients of one variable, since working with
smaller numbers tends to reduce the likelihood of
errors.
4 x 7 28 2 x 3 6
NEXT
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4X 2Y 14 3(4X 2Y 14) 12X 6Y 427X
3Y - 8 2(7X 3Y -8) 14X 6Y -16 26X
0 26 26X 26
X 1Solve for the eliminated variable y
using either of the original equations.4x 2y
144(1) 2y 14 4 2y 14 2y
10 y 5 The solution is (1, 5).
Add the equations to eliminate y.
Start with the given system.
To prepare to eliminate y, multiply the first
equation by 3 and the second equation by 2.
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Practice and Problem Solving

• Solve by elimination
• 1) 2x 5y 17 2) 7x 2y 10
• 6x 5y -9 -7x y -16
• 3) 2x 3y 61 4) 24x 2y 52
• 2x y -7 6x 3y -36
• 5) y 2x 6) 9x 5y 34
• y x 1 8x 2y -2

You choose what method you what to use to solve
question 5 thru 10.
Word Problems
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7) The sum of two numbers is 20. Their
difference is 4. Write and solve a system of
equations.8) Your school sold 456 tickets for
a school play. An adult ticket cost 3.50. A
student ticket cost 1. Total ticket sales
equaled 1131. Let a adult tickets sold, and s
student tickets sold. How many tickets of each
were sold?9) Suppose the band sells cans of
popcorn for 5 each and mixed nuts for 8 each.
The band sells a total of 240 cans and makes a
total of 1614. Find the number of cans of each
sold.
One more problem, my favorite.
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10) A farmer raises chicken and cows. He has a
total of 34 animals in his barnyard. His
six-year son came in one day all excited saying,
Daddy, daddy, did you know all your animals have
a total of 110 legs.Write a system of equations
to represent this situation.How many chickens
and how many cows does the farmer have?
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When you solve systems using elimination, plan a
strategy. The flowchart like the one below can
help you decide how to eliminate a
variable.When you solve systems
using elimination, plan a strategy. Here are a
few hints to help you decide how to eliminate a
variable.

• Answer these questions.

Can I eliminate a variable by adding or
subtracting the given equations?
Do It.
YES Or NO
Can I multiply one of the equations by a number,
and then add or subtract the equations?
YES or NO
Do It.
Multiply both equations by different
numbers. then add or subtract the equations
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