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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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