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Linear Equations
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Terms Involving Equations
3x- 1 = 2
An equation consists of two algebraic expressions joined by an equal sign.
3x1 = 2
3x= 3
x= 1 1 is a solution or root of the equation
Left Side Right Side
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Solve a linear equation 5x - 4 = 7.
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An equation can be transformed into an equivalent equation by one or more of the following
operations.
Example
1. Simplify an expression by
removing grouping symbols andcombining like terms.
3(x - 6) = 6x - x
3x - 18 = 5x
-18 = 2x
-9 = x
Divide both sides of the
equation by 2.
3. Multiply (or divide) on bothsides of the equation by the same
nonzero quantity.
Subtract 3x from both
sides of the equation.
3x - 18 = 5x
3x - 18 - 3x = 5x - 3x
-18 = 2x
2. Add (or subtract) the same
real number or variable
expression on both sides of the
equation.
-9 = x
x = -9
4. Interchange the two sides of
the equation.
Generating Equivalent Equations
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Solve the equation: 2(x- 3) - 17 = 13 - 3(x+ 2).
Solution
Step 1 Simplify the algebraic expression on each side.
2(x- 3)17 = 133(x+ 2) This is the given equation.
2x617 = 133x6 Use the distributive property.
2x23 = - 3x+ 7 Combine like terms.
Text Example
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Solve the equation: 2(x- 3) - 17 = 13 - 3(x+ 2).
Solution
Step 2 Collect variable terms on one side and constant terms on
the other side. We will collect variable terms on the left by adding 3x to
both sides. We will collect the numbers on the right by adding 23 to both
sides.
2x23 + 3x = - 3x+ 7 + 3x Add 3xto both sides.
5x23 = 7 Simplify.
5x23 + 23 = 7 + 23 Add 23 to both sides.
5x= 30 Simplify.
Text Example
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Solve the equation: 2(x- 3) - 17 = 13 - 3(x+ 2).
Solution
Step 3 Isolate the variable and solve. We isolate the variable by
dividing both sides by 5.
5x = 30
5x/5 = 30/5 Divide both sides by 5
x= 6 Simplify.
Text Example
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Solve the equation: 2(x- 3) - 17 = 13 - 3(x+ 2).
Solution
The solution set is {6}.
Step 4 Check the proposed solution in the original equation.
Substitute 6 forxin the original equation.
2(x- 3) - 17 = 13 - 3(x+ 2) This is the original equation.
-11 = -11 This true statement indicates that 6 is the solution.
2(6 - 3) - 17 = 13 - 3(6 + 2) Substitute 6 forx.?
2(3) - 17 = 13 - 3(8) Simplify inside parentheses.?
617 = 1324 Multiply.?
Text Example
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COORDINATE PLANE
Parts of aplane
1. X-axis
2. Y-axis
3. Origin
4. Quadrants
I-IVX-axis
Y-axis
Origin ( 0 , 0)
QUAD IQUAD II
QUAD III QUAD IV
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PLOTTING POINTSRemember when
plotting points you
always start at the
origin. Next you go
left (if x-coordinateis negative) or right
(if x-coordinate is
positive. Then you
go up (if y-coordinate is
positive) or down (if
y-coordinate is
negative)
Plot these 4 points
A (3, -4), B (5, 6), C (-
4, 5) and D (-7, -5)
A
BC
D
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x x2 1
y y2 1
P = ( , )x y1 1
Q = ( , )x y2 2
y
x
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SLOPESlope is the ratio of the vertical rise to the horizontal
run between any two points on a line. Usuallyreferred to as the rise over run.
Slope triangle
between two points.
Notice that the slope
triangle can be drawn
two different ways.Rise is -10
because we
went down
Run is -6because
we went
to the
left
3
5
6
10
iscasethisinslopeThe
Rise is 10because we
went up
Run is 6
because
we went
to the
right
3
5
6
10iscasethisinslopeThe
Another way
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Find the slope of the line between the two points (-4, 8) and (10, -4)
If it helps label the points. 1X 1Y2X 2Y
Then use the
formula
12
12
YY
XX
)8()4(
)4()10(
FORMULAINTOSUBSTITUTE
6
7
12
14
)8(4
410
)8()4(
)4()10(
SimplifyThen
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X AND Y INTERCEPTSThe x-intercept is the x-coordinate of a point
where the graph crosses the x-axis.
The y-intercept is the y-coordinate of a point
where the graph crosses the y-axis.
The x-intercept would be
4 and is located at the
point (4, 0).
The y-intercept is 3
and is located at the
point (0, 3).
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Draw a graph of a line passing throughthe point (1,4) and having a slope -3/2.
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0 5 10
5
5 (1, 4)
(3, 1)-3
2
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The Possibilities for a Lines Slope
Positive Slope
x
y
m > 0
Line rises from left to right.
Zero Slope
x
y
m = 0
Line is horizontal.m is
undefined
Undefined Slope
x
y
Line is vertical.
Negative Slope
x
y
m < 0
Line falls from left to right.
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An equation of a nonvertical line of slope
m that passes through the point ( )x y1 1, is
( )y y m x x 1 1
Point-Slope Equation
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6 4 2 0 2 4 6
6
4
2
2
4
6
(0, 3)
(1, 1)
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Equation of a Horizontal Line
A horizontal line is given by an equation of the form
y=b
Where b is they-intercept.
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Slope-Intercept form of an Equation of a Line
An equation of a lineL with slope m and y-intercept b is
y=mx+b
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Ax By C 0
Th equation of a line L is in general form
when it is written:
WhereA,B, and Care three real numbers and
A andB are not both 0.
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A linear function is a function of the form
f(x)=mx+bThe graph of a linear function is a line with a
slope m and y-intercept b.
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Scatter Diagrams
A relation is a correspondence between two sets. Ifx
andy are two elements and a relation exists betweenx
andy, then we say that xcorresponds toy or that y
depends onx and writex y or we write it as an
ordered pair(x,y).
y - dependent variablex - independent variable
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The first step in finding whether a relation might
exist between two variables is to plot the orderedpairs using rectangular coordinates.
The resulting graph is called a scatter diagram.
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Curve Fitting
Scatter diagrams help us to see the type of relation
that exists between two variables.
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Bivariate data is data in which two variables are
measured on an individual.
The response variable is the variable whose
value can be explained or determined based upon
the value of the predictor variable.
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A scatter diagramshows the relationshipbetween two quantitative variables measured on
the same individual.
Each individual in the data set is represented by apoint in the scatter diagram.
Thepredictor variable is plotted on the horizontal
axis and the response variable is plotted on the
vertical axis.
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Two variables that are linearly related are said to be
positively associated when above average values of
one variable are associated with above average values
of the corresponding variable.
That is, two variables are positively associated when
the values of the predictor variable increase, the values
of the response variable also increase.
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1. The linear correlation coefficient is always between -1
and 1, inclusive. That is, -1 < r< 1.
2. Ifr= +1, there is a perfect positive linear relation
between the two variables.
3. Ifr= -1, there is a perfect negative linear relation
between the two variables.
4. The closerris to +1, the stronger the evidence of
positive association between the two variables.
5. The closerris to -1, the stronger the evidence of
negative association between the two variables.
Properties of the Linear Correlation Coefficient
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6. Ifris close to 0, there is evidence of no linearrelation
between the two variables. Because the linearcorrelation coefficient is a measure of strength of linear
relation, rclose to 0 does not imply no relation, just no
linear relation.
7. It is a unitless measure of association. So, the unit of
measure forx andy plays no role in the interpretation of
r.
Properties of the Linear Correlation Coefficient
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EXAMPLE Drawing a Scatter Diagram and
Computing the Correlation Coefficient
For the following data
(a) Draw a scatter diagram and comment on the type of
relation that appears to exist betweenx andy.
(b) Use technology to compute the linear correlation
coefficient.
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A linear correlation coefficient that implies a
strong positive or negative association that is
computed using observational data does notimply causation among the variables.
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Distinguishing between Linear and
Nonlinear Relations
Lineary=mx+b
m>0
Lineary=mx+b
m
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Nonlinear Nonlinear
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The difference between the observed value
ofyand the predicted value ofyis the error
orresidual. That is
residual = observed - predicted
Compute the residual for the predictioncorresponding to x= 5.
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