Download Graphing Lines and Linear Inequalities, System of Linear Equations

Ordered Pairs
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Graphing Lines and Linear
Inequalities, Solving System of
Linear Equations
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Peter Lo
Ma104 © Peter Lo 2002
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Plotting Points
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All equations in two variables, such as y = mx + c,
is satisfied only if we find a value of x1 and a
value of y 1 that make it true.
These two values are called Ordered Pairs and
denoted as (x1 , y 1 ).
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Cartesian Coordinates System
The Rectangular and Cartesian Coordinates
System consists of a horizontal number line, the
x-axis, and a vertical line, the y-axis.
The intersection of the axes is the Origin.
The axes divide the coordinate plane, or the xyplane, into four regions called the Quadrants.
Locating a point in the rectangular coordinate
system is referred to as plotting or graphing the
point.
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Ma104 © Peter Lo 2002
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Intercepts
Graphing Lines by Plotting Points
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The x-intercept of a line is the point where the
line cross the x-axis.
u The value of x-intercepts is always (x1 , 0)
The y-intercept of a line is the point where the
line cross the y-axis.
u The value of y-intercepts is always (0, y 1 )
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We know that the solution set to an equation is a
line if the equation is linear in two variables – that
is, there are two variables in the problem, and each
variable only appears to a single power, is only in
the numerator of fractions, is never under a radical
symbol, and is not in the same term as the other
variables.
To graph lines by plotting points, first make a
table of values and then plot each point.
Ma104 © Peter Lo 2002
Graphing Lines Using Intercepts
Slope
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Graphing line using intercept is to find the x- and
y-intercepts, and since we know that the equation
represents a straight line, draw a line between
these two points.
This method does not always work if the line
through the origin (0, 0).
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The slope of a line is a measure of its steepness.
A line with slope whose absolute value is very
steep is steep, while a line with a slope whose
absolute value is very steep is shallow.
The formula for the slope m between two points
(x1 , y 1 ) and (x2 , y 2 ) on a line is:
m=
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Ma104 © Peter Lo 2002
y 2 − y1
x 2 − x1
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Slope of Parallel and
Perpendicular Lines
Formation of Equation
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Two lines with slopes m1 and m2 are parallel if
and only if
u m1 = m2
Two lines with slopes m1 and m2 are perpendicular
if and only if
u m1 × m2 = -1
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Information Needed
Point: (x1, y 1)
Point: (x2, y 2)
y − y1 y2 − y1
=
x − x1 x2 − x1
Point-Slope Form
Slope: m
Point: (x1, y 1)
y − y1 = m (x − x1 )
Slope-Intercept
Form
Slope: m
Intercept Form
x-intercept (b, 0)
y-intercept: (0, c)
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Equation
Two Points Form
y-intercept: (0, c)
Ma104 © Peter Lo 2002
Methods for Graphing
a Linear Equation
Comparison of Different Methods
Form
There are four basic methods used to find a
equation of a line from a description of it.
u Two point Form
u Point-Slop Form
u Slope-Intercept Form
u Intercept Form
The equation in the form of Ax + By + C = 0 is
called a Standard Form.
y = mx + c
x y
+ =1
b c
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Linear Inequality
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Graphing a Linear Inequality
If A, B and C are real numbers with A and B are
not zero, then
u Ax + By ≤ C
u Ax + By ≥ C
u Ax + By < C
u Ax + By > C
is called a Linear Inequality.
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Graphing Compound Inequalities
1.
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Example
Graph the lines
u Use a dashed line if the line is not to be included in
the solution set (< or >).
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Use a solid line if the line is to be included in the
solution set ( ≤ or ≥).
The plane of graph is divided into two or more region by
the lines. Find a Test Point for each region.
Check to see if the test point satisfy the inequality.
4.
Shade the appropriate regions.
Graph the solution set to the compound inequality
2x + y ≤ 4 and x – 2y > 7.
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2.
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Ma104 © Peter Lo 2002
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Solving Linear Systems by
Graphing
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Solving Linear Systems using
Substitution
When solving a system of equations by graphing, it is very
important for the graph to be as accurate as possible.
Keep in mind that graph is next exact, and when you solve
by graphing, you need to minimize the errors in the graph.
After solved a system of equations by graph, MUST check
the answer in both of the original equations.
Remember that the point of intersection must lie on both
lines to satisfy both equations.
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2.
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Solve one of the equations for each variables
Substitute this expression into the other equation
(the one that did not use in Step 1)
Solve the resulting equation
Substitute this answer back into your expression
from Step 1
Check the answer in the original equations.
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Example
Solving Linear Systems using the
Addition Method (Elimination Method)
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Solve the system of equations:
3x + 2y = 6
2x - y = 5
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Solve the resulting equation.
Substitute this answer back into either original equation
and solve for the remaining variables.
6.
Check your answer in both of the original equations.
4.
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Write the equations in the same form .
Multiply the equations by a numbers so that in the
resulting equations one of the variables has opposite
coefficients for the two equations.
Add the equation together term-by-term.
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Example
Solving Systems of Linear
Equations in Three Variables
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1.
Solve the system of equations:
3x – 2y = 5
7x + 5y = 2
2.
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Solving Systems of Linear
Equations in Three Variables
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Example
Using the results obtained in Step 1 and Step 2,
form a new system of two equations in two
unknown and solve it using addition or
substitution.
Substitute the value of the two variables obtained
in Step 3 into one of the three original equations
and solve for the value of the third variables.
Check the answer in all three of the original
equation.
Ma104 © Peter Lo 2002
Choose two of the three original equations and,
using addition or substitution, combine the
equations so that one of the three variables is
eliminated.
Choose another pair of the original equation and,
using the same method as in Step 1, combine
these equations to eliminate the same variable as
you did in Step 1.
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Solve the system of equations:
3x + 2y – 3z = 4
x + 3y – 7z = -2
2x – y = 1
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Solving Linear Systems using
Matrices
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Steps for Solving Linear Systems
using Matrices
The augmented matrix and Gaussian elimination can be
used to solve a system of n linear equations in n unknowns
for an natural number n.
To write the augmented matrix, write each equation in
standard form, always keeping the variables lined up
vertically and constants on the right-hand side of the
equation. If a variable is missing, you can either leave the
spot for it empty, of include a zero for it.
Copy the coefficients into the matrix, remembering the
negative signs when appropriate.
The position in the matrix tells which variable the number
is a coefficient of.
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1.
2.
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Interchanging Row (this corresponds to writing
the equations in a different order)
Multiplying a row by a non-zero number (this
corresponds to multiplying one of the equations
by that number)
Replacing a row with a sum of it and another row
(this corresponds to adding two equations)
Ma104 © Peter Lo 2002
Example
Reference
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Solve the system of equations:
2x + y + 2z = 2
x – y – 4z = 3
3x + 2y – 2z = –1
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Algebra for College Students (Ch. 3 – 4)
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