10.1 Straight Lines Concept Overview
STRAIGHT LINES | CONCEPT OVERVIEW The TOPIC of STRAIGHT LINES can be referenced on page 22 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing.
CONCEPT INTRO: The CARTESIAN COORDINATE SYSTEM is a COORDINATE SYSTEM used to represent a position in space based on the RECTANGULAR COORDINATES of each point. The CARTESIAN COORDINATE SYSTEM is based on a GRID, such that every POINT on the PLANE can be identified by UNIQUE x and y coordinates. POSTIONS on the GRID are measured RELATIVE to a FIXED POINT called the ORIGIN, and are MEASURED according to the DISTANCE along a PAIR of AXES.
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The X and Y AXES are treated as a NUMBER LINE, such that POSITIVE DISTANCES are measured to the RIGHT and NEGATIVE to the LEFT for the X AXIS, and POSITIVE DISTANCES are measured UPWARDS and NEGATIVE DOWNWARD for the Y AXIS. Generally, a CARTESIAN COORDINATE SYSTEM can be illustrated with a single RECTANGULAR COORDINATE, (x,y), as such:
The rectangular coordinates are a measure of how far away a point is from a reference axis: β’ The x-coordinate, βπ₯β, is the directed distance from the y-axis to the point of reference β’ The y-coordinate, βπ¦β, is the directed distance from the x-axis to the point of reference.
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The GENERAL FORMULA for the EQUATION OF A STRAIGHT LINE can be referenced under the SUBJECT of MATHEMATICS on page 22 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. The GENERAL FORM of the equation of a STRAIGHT LINE is written as: π΄π₯ + π΅π¦ + πΆ = 0 Where π΄, π΅, and πΆ are real numbers. One example of a general class of equations are linear equations of two variables. The two variables are usually βπ₯β and βπ¦β, but obviously donβt need to be. Linear equations adhere to the following rules: 1. The variables (usually βπ₯β and βπ¦β) appear only to the first power 2. The variables may be multiplied only by real number constants 3. Any real number term may be added (or subtracted) To have a linear equation, all the stated rules must be satisfied, nothing outside of these rules is permitted. ,
As an example, if a stated equation contains elements like x * , y * , , xy, square roots, or -
any other function of βxβ or βyβ, then it is not a straight line. To define a line, we need to specify two distinct pieces of information concerning that line.
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These pieces of information could be two distinct points that the line passes through, or a single point that it passes through along with some representation of itβs βslopeβ. The GENERAL FORMULA representing the SLOPE OF A LINE can be referenced under the SUBJECT of MATHEMATICS on page 22 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. The SLOPE OF A LINE is a measure of how βtiltedβ the line is and can be equated using the standard formula:
π=
πππ π (π¦* β π¦, ) = ππ’π (π₯* β π₯, )
Where: β’ (π₯, , π¦, ) are the βxβ and βyβ coordinates of point 1 β’
π₯* , π¦* are the βxβ and βyβ coordinates of point 2
It makes no difference what two points are used for point 1 and point 2. If they were flipped in our general formula, both the numerator and the denominator of the fraction would be changed to the opposite sign, giving exactly the same result.
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GENERAL TIP: The slope of the line is also the derivative of the straight line equation given. For example, say we are given a straight line represented by the following formula: π¦ = 4π₯ Taking the derivative of this equation will give us: π¦ H = 4 The result is a whole, real number of 4, this is the slope.
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SLOPE-INTERCEPT FORM:
The FORMULA for the SLOPE-INTERCEPT FORM OF A STRAIGHT LINE can be referenced under the SUBJECT of MATHEMATICS on page 22 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. The POINT representing the INTERSECTION of a STRAIGHT LINE and a COORDINATE AXIS is known as an INTERCEPT. The TWO INTERCEPTS we will be working with are the X-INTERCEPT and the YINTERCEPT.
The X-INTERCEPT (π) is the POINT on the LINE where Y is ZERO, such that the point can be written as π, 0 .
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The Y-INTERCEPT (b) is the POINT on the LINE where X is ZERO, such that the point can be written as (0, b). The STANDARD FORM of a STRAIGHT LINE is also known as the SLOPEINTERCEPT FORM: π¦ = ππ₯ + π Where: β’ π is the slope β’ π is the y-intercept β’
π₯, π¦ are coordinates
Where βπβ and βπβ are any two real numbers, and defines the straight line with a slope of βπβ and a βπ¦β intercept equal to βπβ.
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POINT-SLOPE FORM: The FORMULA for the POINT-SLOPE FORM OF A STRAIGHT LINE can be referenced under the topic of STRAIGHT LINES on page 22 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. Defining the point-slope formula of a straight line we have: π¦ β π¦, = π(π₯ β π₯, ) A line can be defined using the point-slope formula if one distinct point and the slope of the line are given. TWO-POINT FORM: Another way to define a line is to identify two distinct points that the line passes through. If it is given that a line passes through the points (π₯, , π¦, ) and (π₯* , π¦* ) then the two-point formula can be stated as: π¦ β π¦, = π π₯ β π₯, Plugging in what we know about the slope, as defined above, we have:
π¦ β π¦, =
π¦* β π¦, (π₯ β π₯, ) π₯* β π₯, Made with
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The FORMULA allowing us to determine the ANGLE BETWEEN LINES can be referenced under the SUBJECT of MATHEMATICS on page 22 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. The angle between two distinct lines with slopes π, and π* can be determined using that formula:
πΌ = arctan
π* β π, 1 + π* π,
GENERAL REMINDER: arctan = π‘ππN,
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TYPES OF LINES: The FORMULA for the SLOPE OF A PERPENDICULAR LINE can be referenced under the SUBJECT of MATHEMATICS on page 22 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. PARRALEL LINES are LINES that have the SAME SLOPE such they NEVER INTERSECT.
Two lines are considered PARALLEL if they have the same slopes such that: π, = π*
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PERPENDICULAR LINES are LINES that have SLOPES that are NEGATIVE RECIPROCALS. Two lines are considered PERPENDICULAR if their slopes are inversely proportional such that:
π, = β
1 π*
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A line is considered HORIZONTAL when it shares the same y-coordinate on more than one position in space, which would make:
π=
πππ π =0 ππ’π
A line is considered VERTICAL when it shares the same x-coordinate on more than one position in space. A vertical line has an undefined slope due to: π₯, = π₯*
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And because the slope of a line is defined by:
π=
πππ π π¦* β π¦, = ππ’π π₯* β π₯,
Having the two equivalent x elements creates a zero value in the denominator and further, an undefined slope.
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STRAIGHT LINES | CONCEPT EXAMPLE The following problem introduces the concept reviewed within this module. Use this content as a primer for the subsequent material. The equation representing the line, in slope-intercept form, that passes through the points (β1,2) and (2,1) is most close to: A. y = x β 20 ,
B. y = β x β 5 O
,
C. y = x + 5 Q
,
Q
R
R
D. y = β x +
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SOLUTION: The TOPIC of STRAIGHT LINES can be referenced under the SUBJECT of MATHEMATICS on page 22 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. Recall that to define a line we need to specify two distinct pieces of information. In this case, we have two distinct points given, which is sufficient in allowing for us to define the line that runs through them. Letβs revisit the general slope-intercept formula, which is: π¦ = ππ₯ + π Where: β’ π is the slope β’ π is the y-intercept β’
π₯, π¦ are coordinates
The FORMULA for the SLOPE-INTERCEPT FORM OF A STRAIGHT LINE can be referenced under the SUBJECT of MATHEMATICS on page 22 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing.
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The first step is to define the slope, which is given by the formula:
π=
πππ π (π¦* β π¦, ) = ππ’π (π₯* β π₯, )
Where: β’ (x, , y, ) are the x and y coordinates of point 1 β’
x* , y* are the x and y coordinates of point 2
Given the two points as (β1,2) and (2,1), we simply plug and play to get:
π=
(1 β 2) 1 =β (2 β β1 ) 3
With the slope now defined, the last characteristic we need to define in the yintercept. We can do this by taking one of the points and plugging it in to the general slopeintercept equation: π¦ = ππ₯ + π Where again: β’ π is the slope β’ π is the y-intercept
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We can rearrange and isolate the π¦-intercept, βπβ such that: π = π¦, β ππ₯, We have defined the slope, and we also have the point β1, 2 , which allows us to plug and chug to define the y-intercept:
π =2β β
1 3
β1 =
5 3
With both the slope βπβ and the y-intercept βπβ defined, we can place them in to general slope-intercept form and define the formula for the line as: 1 5 π¦=β π₯+ 3 3 π
π
π
π
Therefore, the correct answer choice is D. π = β π +
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CONCEPT INTRO: The CARTESIAN COORDINATE SYSTEM is a COORDINATE SYSTEM used to represent a position in space based on the RECTANGULAR COORDINATES of each point. The CARTESIAN COORDINATE SYSTEM is based on a GRID, such that every POINT on the PLANE can be identified by UNIQUE x and y coordinates. POSTIONS on the GRID are measured RELATIVE to a FIXED POINT called the ORIGIN, and are MEASURED according to the DISTANCE along a PAIR of AXES.
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The X and Y AXES are treated as a NUMBER LINE, such that POSITIVE DISTANCES are measured to the RIGHT and NEGATIVE to the LEFT for the X AXIS, and POSITIVE DISTANCES are measured UPWARDS and NEGATIVE DOWNWARD for the Y AXIS. Generally, a CARTESIAN COORDINATE SYSTEM can be illustrated with a single RECTANGULAR COORDINATE, (x,y), as such:
The rectangular coordinates are a measure of how far away a point is from a reference axis: β’ The x-coordinate, βπ₯β, is the directed distance from the y-axis to the point of reference β’ The y-coordinate, βπ¦β, is the directed distance from the x-axis to the point of reference.
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The GENERAL FORMULA for the EQUATION OF A STRAIGHT LINE can be referenced under the SUBJECT of MATHEMATICS on page 22 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. The GENERAL FORM of the equation of a STRAIGHT LINE is written as: π΄π₯ + π΅π¦ + πΆ = 0 Where π΄, π΅, and πΆ are real numbers. One example of a general class of equations are linear equations of two variables. The two variables are usually βπ₯β and βπ¦β, but obviously donβt need to be. Linear equations adhere to the following rules: 1. The variables (usually βπ₯β and βπ¦β) appear only to the first power 2. The variables may be multiplied only by real number constants 3. Any real number term may be added (or subtracted) To have a linear equation, all the stated rules must be satisfied, nothing outside of these rules is permitted. ,
As an example, if a stated equation contains elements like x * , y * , , xy, square roots, or -
any other function of βxβ or βyβ, then it is not a straight line. To define a line, we need to specify two distinct pieces of information concerning that line.
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by Prepineer | Prepineer.com
These pieces of information could be two distinct points that the line passes through, or a single point that it passes through along with some representation of itβs βslopeβ. The GENERAL FORMULA representing the SLOPE OF A LINE can be referenced under the SUBJECT of MATHEMATICS on page 22 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. The SLOPE OF A LINE is a measure of how βtiltedβ the line is and can be equated using the standard formula:
π=
πππ π (π¦* β π¦, ) = ππ’π (π₯* β π₯, )
Where: β’ (π₯, , π¦, ) are the βxβ and βyβ coordinates of point 1 β’
π₯* , π¦* are the βxβ and βyβ coordinates of point 2
It makes no difference what two points are used for point 1 and point 2. If they were flipped in our general formula, both the numerator and the denominator of the fraction would be changed to the opposite sign, giving exactly the same result.
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GENERAL TIP: The slope of the line is also the derivative of the straight line equation given. For example, say we are given a straight line represented by the following formula: π¦ = 4π₯ Taking the derivative of this equation will give us: π¦ H = 4 The result is a whole, real number of 4, this is the slope.
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SLOPE-INTERCEPT FORM:
The FORMULA for the SLOPE-INTERCEPT FORM OF A STRAIGHT LINE can be referenced under the SUBJECT of MATHEMATICS on page 22 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. The POINT representing the INTERSECTION of a STRAIGHT LINE and a COORDINATE AXIS is known as an INTERCEPT. The TWO INTERCEPTS we will be working with are the X-INTERCEPT and the YINTERCEPT.
The X-INTERCEPT (π) is the POINT on the LINE where Y is ZERO, such that the point can be written as π, 0 .
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The Y-INTERCEPT (b) is the POINT on the LINE where X is ZERO, such that the point can be written as (0, b). The STANDARD FORM of a STRAIGHT LINE is also known as the SLOPEINTERCEPT FORM: π¦ = ππ₯ + π Where: β’ π is the slope β’ π is the y-intercept β’
π₯, π¦ are coordinates
Where βπβ and βπβ are any two real numbers, and defines the straight line with a slope of βπβ and a βπ¦β intercept equal to βπβ.
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POINT-SLOPE FORM: The FORMULA for the POINT-SLOPE FORM OF A STRAIGHT LINE can be referenced under the topic of STRAIGHT LINES on page 22 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. Defining the point-slope formula of a straight line we have: π¦ β π¦, = π(π₯ β π₯, ) A line can be defined using the point-slope formula if one distinct point and the slope of the line are given. TWO-POINT FORM: Another way to define a line is to identify two distinct points that the line passes through. If it is given that a line passes through the points (π₯, , π¦, ) and (π₯* , π¦* ) then the two-point formula can be stated as: π¦ β π¦, = π π₯ β π₯, Plugging in what we know about the slope, as defined above, we have:
π¦ β π¦, =
π¦* β π¦, (π₯ β π₯, ) π₯* β π₯, Made with
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The FORMULA allowing us to determine the ANGLE BETWEEN LINES can be referenced under the SUBJECT of MATHEMATICS on page 22 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. The angle between two distinct lines with slopes π, and π* can be determined using that formula:
πΌ = arctan
π* β π, 1 + π* π,
GENERAL REMINDER: arctan = π‘ππN,
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TYPES OF LINES: The FORMULA for the SLOPE OF A PERPENDICULAR LINE can be referenced under the SUBJECT of MATHEMATICS on page 22 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. PARRALEL LINES are LINES that have the SAME SLOPE such they NEVER INTERSECT.
Two lines are considered PARALLEL if they have the same slopes such that: π, = π*
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PERPENDICULAR LINES are LINES that have SLOPES that are NEGATIVE RECIPROCALS. Two lines are considered PERPENDICULAR if their slopes are inversely proportional such that:
π, = β
1 π*
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A line is considered HORIZONTAL when it shares the same y-coordinate on more than one position in space, which would make:
π=
πππ π =0 ππ’π
A line is considered VERTICAL when it shares the same x-coordinate on more than one position in space. A vertical line has an undefined slope due to: π₯, = π₯*
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And because the slope of a line is defined by:
π=
πππ π π¦* β π¦, = ππ’π π₯* β π₯,
Having the two equivalent x elements creates a zero value in the denominator and further, an undefined slope.
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STRAIGHT LINES | CONCEPT EXAMPLE The following problem introduces the concept reviewed within this module. Use this content as a primer for the subsequent material. The equation representing the line, in slope-intercept form, that passes through the points (β1,2) and (2,1) is most close to: A. y = x β 20 ,
B. y = β x β 5 O
,
C. y = x + 5 Q
,
Q
R
R
D. y = β x +
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SOLUTION: The TOPIC of STRAIGHT LINES can be referenced under the SUBJECT of MATHEMATICS on page 22 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. Recall that to define a line we need to specify two distinct pieces of information. In this case, we have two distinct points given, which is sufficient in allowing for us to define the line that runs through them. Letβs revisit the general slope-intercept formula, which is: π¦ = ππ₯ + π Where: β’ π is the slope β’ π is the y-intercept β’
π₯, π¦ are coordinates
The FORMULA for the SLOPE-INTERCEPT FORM OF A STRAIGHT LINE can be referenced under the SUBJECT of MATHEMATICS on page 22 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing.
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The first step is to define the slope, which is given by the formula:
π=
πππ π (π¦* β π¦, ) = ππ’π (π₯* β π₯, )
Where: β’ (x, , y, ) are the x and y coordinates of point 1 β’
x* , y* are the x and y coordinates of point 2
Given the two points as (β1,2) and (2,1), we simply plug and play to get:
π=
(1 β 2) 1 =β (2 β β1 ) 3
With the slope now defined, the last characteristic we need to define in the yintercept. We can do this by taking one of the points and plugging it in to the general slopeintercept equation: π¦ = ππ₯ + π Where again: β’ π is the slope β’ π is the y-intercept
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We can rearrange and isolate the π¦-intercept, βπβ such that: π = π¦, β ππ₯, We have defined the slope, and we also have the point β1, 2 , which allows us to plug and chug to define the y-intercept:
π =2β β
1 3
β1 =
5 3
With both the slope βπβ and the y-intercept βπβ defined, we can place them in to general slope-intercept form and define the formula for the line as: 1 5 π¦=β π₯+ 3 3 π
π
π
π
Therefore, the correct answer choice is D. π = β π +
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