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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?