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12.1 Solving Systems of
EquationsSolve systems of equations by graphing, substitution, and elimination.
Recognize consistent and inconsistent systems.
Solve applications using systems.
Solving Systems of Equations System of equations:
3 x 2 has _____________________
3 x 3 has _____________________
3 x 4 has _____________________
2 x 2 has ______________________
What type of systems are below?
2x – 3y + 4z = 11 2x2 + y = 10
x + 4y – 5z = 7 x2 – y2 = 5
3x – 8y + 9z = 22
7x + 2y – 3z + w = 6
x + 4y + 5z – 2w = 11
9x – 8y + 4z + w = 18
Solving Systems of Equations
Determine whether the given values of x, y,
and z are a solution of the system of
equations: x = 1, y = 2 z = 3
2x – 5y + 3z = 1
x + 2y – z = 2
3x + y + 2z = 11
Same system but x = 0, y = 7 z = 12
Solutions of a System of Equations
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Solving a system graphically
Write the equation in slope-intercept form.
◦ (where y is by itself)
Put both equations in the graphing calculator
Find the point of intersection-menu, analyze,
intersection, lower bound (left of the
intersection point), enter, upper bound (right
of the intersection point), enter.
2x – y = 1
3x + 2y = 12
Find a solution of the system of equations
below by graphing the equations.
2x – y = 1
3x + 2y = 4
x + 2y =5
2x – 3y = 7
Solving a system graphically
Types of System and Number of Solutions
Yellow box page 782
y
x
y
x
Lines intersectone solution
Consistent system
Lines are parallelno solution
Inconsistent system
y
x
Lines coincideinfinitely many solutions
Consistent system
Solving Systems with SubstitutionSTEP 1: Solve one of the equations for one of
its variables. (Get a letter by itself)
STEP 2: Substitute the expression from Step 1 into the other equation & solve. (ONLY have 1 variable now)
STEP3: Substitute the value from Step 2 into the revised equation in step 1 & solve.
STEP4: Check the solution, (ordered pair) in each of the original equations.
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3x – y = 12
2x + 3y = 2
Solving Systems with Substitution
2x + y = 2
3x – 2y = 4
The Elimination Method STEP 1 – Multiply one or both of the equations
by a constant to obtain the coefficients that differ
only in sign for one of the variables.
STEP 2 – Add the revised equations from Step 1.
Combining like terms will eliminate one of the
variables. Solve for the remaining variables.
STEP 3 – Substitute the value obtained in Step 2
into either of the original equations and solve for
the other variable.
Solving a system by elimination Solve the system of equatons below by
elimination
x – 3y = 4
2x + y = 1
Solving a system by elimination Solve the system of equatons below by
elimination
3x + y = 5
x – 2y = 4
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Solutions of Consistent and
Inconsistent Systems An inconsistent system- the variable cancels
out and is false ex: 0 = 4. The lines are
parallel.
A consistent system- the variables cancel
out and is true ex: 4 = 4. The lines coincide.
(When the variables don’t cancel out there
is only one solution. The lines intersect.)
2x – 3y = 5 2x – 4y = 6
4x – 6y = 1 -3x + 6y = -9
3x – 6y = 3 3x – 2y = 7
-4x + 8y = -4 9x – 6y = -3
Solutions of Consistent and
Inconsistent Systems
Solving a 3 x 3 System by Elimination
Eliminate one variable in the one pair of equations and then in another pair so a 2 x 2 remains.
Follow the same steps as elimination for a 2 x 2.
Solve for the remaining variable.
Solve the system of equations below by elimination.2x + y – z = -1-x – 3y + z = 5x + 4y – 2z = -10
Solve the system of equations below by
elimination.
x + y – z = 6
2x + 3y + z = 5
-x + 2y + 4z = -9
Solving a 3 x 3 System by
Elimination
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Applications of Systems
A ball game is attended by 575 people, and
total ticket sales are $2575. If tickets cost
$5 for adults and $3 for children, how many
adults and how many children attended the
game?
Applications of Systems
The revenue from ticket sales for a concert
was $7050, and 910 tickets were sold.
Reserved seat tickets cost $9 and lawn seat
tickets cost $5. How many of each type of
ticket were sold?
Applications of Systems
A café sells two kinds of coffee in bulk. the
Costa Rican sells for $4.50 per pound, and
the Kenyan sells for $7.00 per pound. The
owner wishes to mix a blend that would sell
for $5.00 per pound. How much of each
type of coffee should he used in the blend?
Applications of Systems
A market sells macadamia nuts for $8.00 a
pound and almonds for $5.50 a pound. A
customer wishes to make a mix that would
cost $7.00 a pound. How much of each type
of nut will be used in a pound of mix?
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