LINEAR AND QUADRATIC EQUATION SYSTEMS

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LINEAR AND QUADRATIC EQUATION SYSTEMS

• In this chapter, you will learn how to
• solve two simultaneous where at least one is a
  non-linear equation
• solve three simultaneous linear equations

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Linear and quadratic equation systems
Linear equation systems in two variable
Simultaneous Equations Two Quadratics
Linear equation systems in three variables
Applications
Exercises
Simultaneous equations one linear and one
quadratic
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Linear equations system in two variables

• We have seen in Junior High school Mathematics
  that a set of simultaneous linear equations can
  be solved by either the method of
• Elimination
• Substitution
• Combination both elimination and substitution

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In this lesson you will study two algebraic
methods for solving linear systems. The first
method is called substitution.
Solve one of the equations for one of its
variables.
Substitute expression from Step 1 into other
equation and solve for other variable.
Substitute value from Step 2 into revised
equation from Step 1. Solve.
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3 x 4y ? 4 Equation 1 x 2y ? 2
Equation 2
Solve the linear system using the substitution
method.
SOLUTION
Solve Equation 2 for x.
x 2y ? 2
Write Equation 2.
x ? 2y 2
Revised Equation 2.
Substitute the expression for x into Equation 1
and solve for y.
3x 4y ? 4
Write Equation 1.
Substitute 2y 2 for x.
3( 2y 2) 4y ? 4
y ? 5
Simplify.
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3 x 4y ? 4 Equation 1 x 2y ? 2
Equation 2
Solve the linear system using the substitution
method.
Substitute the value of y into revised Equation 2
and solve for x.
x ? 2y 2
Write revised Equation 2.
Substitute 5 for y.
x ? 2(5) 2
x ? 8
Simplify.
The solution is ( 8, 5).
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3 x 4y ? 4 Equation 1 x 2y ? 2
Equation 2
Solve the linear system using the substitution
method.
Check the solution by substituting back into the
original equation.
3x 4y ? 4
x 2y ? 2
Write original equations.
Substitute x and y.
Solution checks.
4 ? 4
2 ? 2
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CHOOSING A METHOD In the first step of the
previous example, you could have solved for
either x or y in either Equation 1 or Equation 2.
It was easiest to solve for x in Equation 2
because the x-coefficient was 1. In general you
should solve for a variable whose coefficient is
1 or 1.
If neither variable has a coefficient of 1 or 1,
you can still use substitution. In such cases,
however, the linear combination method may be
better. The goal of this method is to add the
equations to obtain an equation in one variable.
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Multiply one or both equations by a constant to
obtain coefficients that d iffer only in sign
for one of the variables.
Add revised eq uations from Step 1. Combine like
terms to eliminate one of the variables. Solve
for remaining variable.
Substitute value obtained in Step 2 into either
original equation and solve for other variable.
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Solve the linear system using thelinear
combination method.
2 x 4y ? 13 Equation 1 4 x 5y ? 8 Equation 2
SOLUTION
Multiply the first equation by 2 so that
x-coefficients differ only in sign.
2
4x 8y ? 26
2 x 4y ? 13
4 x 5y ? 8
4 x 5y ? 8
3y ? 18
Add the revised equations and solve for y.
y ? 6
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Solve the linear system using thelinear
combination method.
2 x 4y ? 13 Equation 1 4 x 5y ? 8 Equation 2
Substitute the value of y into one of the
original equations.
2 x 4y ? 13
Write Equation 1.
Substitute 6 for y.
2 x 4( 6) ? 13
2 x 24 ? 13
Simplify.
Solve for x.
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7 x 12 y ? 22 Equation 1 5 x 8 y ?
14 Equation 2
Solve the linear system using thelinear
combination method.
SOLUTION
Multiply the first equation by 2 and the second
equation by 3 so that the coefficients of y
differ only in sign.
2
7 x 12 y ? 22
14 x 24y ? 44
15 x 24y ? 42
3
5 x 8 y ? 14
x ? 2
Add the revised equations and solve for x.
x ? 2
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7 x 12 y ? 22 Equation 1 5 x 8 y ?
14 Equation 2
Solve the linear system using thelinear
combination method.
Substitute the value of x into one of the
original equations. Solve for y.
5 x 8 y ? 14
Write Equation 2.
Substitute 2 for x.
5 (2) 8 y ? 14
y 3
Solve for y.
The solution is (2, 3).
Check the solution algebraically or graphically.
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x 2 y ? 3 2 x 4 y ? 7
Solve the linear system
SOLUTION
Since the coefficient of x in the first equation
is 1, use substitution.
Solve the first equation for x.
x 2 y ? 3
x ? 2 y 3
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x 2 y ? 3 2 x 4 y ? 7
Solve the linear system
Substitute the expression for x into the second
equation.
2 x 4 y ? 7
Write second equation.
2(2 y 3) 4 y ? 7
Substitute 2 y 3 for x.
6 ? 7
Simplify.
Because the statement 6 7 is never true, there
is no solution.
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6 x 10 y ? 12 15 x 25 y ? 30
Solve the linear system
SOLUTION
Since no coefficient is 1 or 1, use the linear
combination method.
30 x 50 y ? 60
5
6 x 10 y ? 12
30 x 50 y ? 60
2
15 x 25 y ? 30
0 ? 0
Add the revised equations.
Because the equation 0 0 is always true, there
are infinitely many solutions.
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Exercise 1.
Work in pair and answer these questions

• Solve each system
• Translate to a system of equations and solve The
  sum of two numbers is 82. One is twelve more than
  the other. Find the larger number.

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Exercise 1

• 3. Translate to a system of equations and solve
  Alfa Rent-a-Car rents compact cars at daily rate
  of 23.95 plus 20 per kilometer. Giant
  Rent-a-Car rents compact cars at daily rate of
  22.95 plus 22 per kilometer. For what kilometer
  is cost will be same?

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Exercise 1

• 4. Abu works for 6 days which 4 days are overtime
  to get Rp74.000,00. Budi works for 5 days which 2
  days are overtime to get Rp55.000,00. Abu, Budi,
  and Catur work under the same payment system.
  Catur works for 5 days overtime. Find the payment
  that he will receive.

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Linear equation systems in three variables

• General form of linear equation systems in three
  x, y, z
• variables is
• whereas

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Answering the linear equation system with
three variables is determining
that is a
simultaneous solution from the
equation system, so the solution set is
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Example 1

• Find the solution set of
• Solution
• 
  
• Substitute to

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• Then we will find the linear equation system
  
• 
  
• And it is substituted into 5 x 8 y 4 4 we
  will find

And put the result into
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Exercise 2

• Solve this system
• Three lines 3x-y10, 2x-y-30, and
  x-ay-70 are concurrent. Find the value of a
• The price of 2 guavas, 2 bananas, and a mango are
  Rp1400,00. The price of a guava, a banana, and 2
  mangoes are Rp1300,00. The price of 3 guavas, a
  banana, and a mango are Rp1500,00. Find the price
  when you buy a guava, a banana, and a mango.

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Exercise 2

• 4. Do the exercises in p. . no., p. .
  number

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Simultaneous equations, one linear and one
quadratic

• General form of this system is
• where

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Example
Solve the following system
(1)

(2)
Solution From the linear equation y 3x-1
substitute to (2)
?(4x-1)(x-2)0
? x ¼ or x 2
The solution is either x ¼ gt y - ¼ or x
2gt y 5
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Exercise 3

1. Solve the following system
2. Find the intersection of line y 6x-26 and
   parabola and sketch.
3. Find tangent to the parabola at the
   point A(3,9), and sketch
4. The line m intersects the circle
   at A(3,4) and B. Find coordinate of point B.

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Simultaneous Equations Two Quadratics

• Solution of system
• can be found by
• or
• after x has been found from the last quadratic
  equation then the value of y can be obtained by
  substituting the value of x into one of
  equations in the system

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• Find the intersection of parabolas
• and

Example
Solution
? x 3 or x 1
When x 3, y 13, and x 1, y - 1
The intersection are (3,13) and (1, - 1)
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Exercise 4

• 1.Find the value of m so that the line y mx 2
  touches the parabola ,
  also find the coordinate of intersection.
• 2. If x and y are acute angles that satisfy the
  following system
• find the angles x and y in degrees.

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Exercise 1

• Exercise 2

which one do you want to try ?
Exercise 3
Exercise 4
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Simultaneous equations one is linear and the
other one is quadratic

• General form of this system is
• whereas