Algebra Module A46 Linear Equation Theor - Copright This publication The Northern Alberta Institute of Technolog 00. All Rights Reserved. LAST REVISED June., 009
Linear Equation Theor - Statement of Prerequisite Skills Complete all previous TLM modules before beginning this module. Required Supporting Materials Access to the World Wide Web. Internet Eplorer. or greater. Macromedia Flash Plaer. Rationale Wh is it important for ou to learn this material? This module epands on the knowledge gained in the linear equation theor module. The skills gained in this module will allow ou to solve problems involving straight lines when onl partial information about the problem is given. Learning Outcome When ou complete this module ou will be able to Solve problems involving straight lines. Learning Objectives. Change a given linear equation into an equivalent form.. Determine the equation of a line using the point slope form of a line, given two points on the line or given one point and the slope of the line. 3. Determine the slope of an oblique line given its equation. 4. Determine the equation of a line given its graph.. Determine the slope, intercept, intercept, and/or the equation of a horizontal or vertical line. 6. Appl the relationships that occur between lines that are parallel and those that occur between lines that are perpendicular. 7. Solve word problems relating to straight lines. Connection Activit Consider a triangle with vertices A( 3, 4), B(4, ), C(, ). What would ou do to determine the length from point A to Point C? Module A46 Linear Equation Theor -
OBJECTIVE ONE When ou complete this objective ou will be able to Change a given linear equation into an equivalent form. Eploration Activit If calculators or computers are to be used in solving man questions in coordinate geometr, it is often necessar to place epressions in a standardized format. For eample the line = + and the line + = represent the same relationship between and. This objective reviews the manipulation of algebraic epressions into standardized form.. Standard form A + B = C A, B, C are integers and A 0. intercept form (also called slope/intercept form) = m + b m and b are an real numbers where m is the slope of the line and b represents the intercept 3. Point-slope form Comment: l = m( l ), l, and m are an real numbers where m is the slope of the line and (, l ) represents a point on the line. The student should be aware that the slope/intercept form (form above) is ver useful as a quick wa of determining the slope of the line. This is possible because the coefficient of in this form is the slope. The student must be careful though, to make sure the coefficient of is + when using this form of the equation of a straight line.. Form (Standard) is the form we most often start or finish with. 3. Form 3 (point-slope) will be dealt with in detail in Objective. It is a ver useful form. Module A46 Linear Equation Theor -
EXAMPLE Epress = 3( 4) in standard form and state the values of A, B, and C. SOLUTION: = 3( 4) Remove braces to get: = 3 Rearrange terms: 3 + = 7 Multipl through b - to get: 3 = 7 Compare 3 = 7 with A + B = C then we see that: A = 3, B = and C = 7 It is important to understand where values for A, B, and C come from when entering answers into TLM. Note: You must rearrange the equation so that A is positive!!! EXAMPLE Epress 3 7 = 0 in intercept form and state the values of m and b. SOLUTION: 3 7 = 0 = 3 + 7 = 3 7 Compare = 3 7 with = m + b and see that: m = 3 and b = 7 It is important to understand where values for m, and b come from when entering answers into TLM. Module A46 Linear Equation Theor - 3
Eperiential Activit One. Epress 4 + 7 = 0 in the form = m + b.. Epress 6 + = 7 in the form = m + b. 3. Epress = 3 in the form A + B = C. 4. Epress = in the form A + B = C. 3. Epress 3 = ( + 4) the form A + B = C. Show Me. 3 6. Epress + 4 = in = m + b form. 7. Epress = in the form A + B = C. 8. Epress = + in the form A + B = C and state values for A, B, C. 3 9. Epress 7 4 + = 0 in the form = m + b and state values for m, b.. Epress = + in the form A + B = C and state values for a, b, c. 4 4 Module A46 Linear Equation Theor -
Eperiential Activit One Answers. = 4 + 7. 7 = 3 + 3. + = 3 4. 3 =. 3 = 7 6. = + 4 7. = 8. A = B = 3 C = 9. 7 m = 4 b =. A = B = 4 C = 8 Module A46 Linear Equation Theor -
OBJECTIVE TWO When ou complete this objective ou will be able to Determine the equation of a line using the point-slope form of the line, given two points on the line or given one point and the slope of the line. Eploration Activit Point - Slope Equation To get the Point Slope equation, we start with the slope formula: m = The slope formula uses two specific points on a line. If we replace (, ) with the general point (, ) we get: m = which simplifies to: = m( ) Equation () This is the Point Slop Equation!! Equation () is the point-slope form of the equation of a line and is ver important in an stud of linear equations. What it means is that given an two points on the line or given one point and the slope of the line we can solve for the equation of the line. EXAMPLE Find the standard equation of the line that passes through the point (6, 0) and that has a slope of 4. SOLUTION: Since m = 4 and ( l ) = (6, 0) we can substitute directl into equation () above: ( 0) = ( 6) 4 + 0 = ( 6) multipl both sides b 4 4 4 + 80 = 6 + 4 = 86 we generall make the coefficient positive, so 4 = 86 Thus 4 = 86 is the equation of the line. 6 Module A46 Linear Equation Theor -
EXAMPLE Determine the equation of the line passing through the points (, 6) and (8, 40). SOLUTION: First we must find the slope 6 40 m = 8 34 = 9 and then taking either point as (, l ) and the slope, we substitute into (), and get: 34 40 = ( 8) 9 9 360 = 34 7 34 + 9 = 88 34 9 = 88 the equation of the line. NOTE: The student should realize that being given points on the line is essentiall the same thing as being given one point and the slope. Module A46 Linear Equation Theor - 7
Eperiential Activit Two Determine the equations of the following straight lines in standard form:. Passing through (4, 6) and (, ). Show Me.. Passing through ( 0, 0) and ( 0, 8). 3. Passing through (, 30), and (, 40). 4. Passing through (6, ) with slope equal to 3.. Passing through ( 8, 0) with slope equal to 6. 6. Passing through (80, ) with slope equal to 7. Passing through (,7) and (, 9). 8. Passing through ( 3, 7) and ( 8, 3).. 9. Passing through ( 4, 9) with slope equal to 3.. Passing through (, 7) with slope equal to. 3 Eperiential Activit Two Answers. 8 3 = 0. 4 = 400 3. 3 + 3 = 6 4. 3 =. 6 + = 8 6. + = 78 7. + 7 = 8. 6 = 7 9. 3 = 33. + 3 = 6 8 Module A46 Linear Equation Theor -
OBJECTIVE THREE When ou complete this objective ou will be able to Determine the slope of an oblique line given its equation. Eploration Activit EXAMPLE Determine slope of the line give represented b the equation 3 = SOLUTION: METHOD Find an two points on the line and then appl the slope formula. Take = then 3() = 6 = = = So, is a point, let's call it (, ) Now take = 0 then 3(0) = = = So 0, is another point, let's call it (, ) Module A46 Linear Equation Theor - 9
Now appl the slope formula m = m = 0 3 m = the slope is 3 which means the line rises 3 units for ever units it moves horizontall. METHOD Use the intercept form of the equation of the line to determine the slope of a line. Change the given equation 3 = into the form = m + b where the coefficient of represents the slope of the line. To do this solve for : 3 = = 3 + 3 3 = + = 3 Now compare = with = m + b and see here that: m = 3 is the slope, and b = is the - intercept. Note: For this method to work the coefficient of must alwas be +. For slope questions obtained from the computer based question bank alwas reduce the fraction to lowest terms before identifing rise and run. Also put rise as + or, and put run as +. COMMENT: Of the two methods just presented, Method is b far the more efficient wa to go. Module A46 Linear Equation Theor -
Eperiential Activit Three. Determine the slope of the line described b 8 = 3. Show Me.. Determine the slope of the line described b 6 + =. 3. Determine the slope of the line described b = 3 + 6. 4. Determine the slope of the line described b =. Determine the slope of the line described b + = 80. 6. Determine the slope of the line described b 8 4 = 9. 7. Determine the slope of the line described b 3 + = 7. 8. Determine the slope of the line described b 7 + 3 =. Eperiential Activit Three Answers.. 3. 4. 8 6 3. 6. 7. 8. 4 7 3 Module A46 Linear Equation Theor -
OBJECTIVE FOUR When ou complete this objective ou will be able to Determine the equation of a line given its graph. Eploration Activit EXAMPLE Given: 6 4 p 3 p (, ) (, ) -6-4 - p 4 6 (, 0) - -4-6 FIND: The slope of the given line. METHOD SOLUTION: Given p l is (, 0) and p is (, ), then we use the slope formula to get: slope of p p = m = m = 0 Module A46 Linear Equation Theor -
Now let us pick a general point (, ) on the line and substitute it into the slope formula for (, ) to obtain: m = = ( ) = ( ) = 4 = the desired equation in Standard form. METHOD Appl the formula: l = m( l ) Given: p = (, 0) here = and = 0 p = (, ) here = and = To find the slope we use the formula m = so the slope p p = 0 = m = Substitute m = and (, ) = (, 0) into the formula = m( ): 0 = ( ) = + = = Standard form. NOTE: This method is the more commonl used method. Module A46 Linear Equation Theor - 3
Eperiential Activit Four Epress answers in the form indicated. 6 4 l -6-4 - 4 6 - l 3-4 -6 l. Determine the equation of l in standard form. Show Me.. Determine the equation of l in standard form. 3. Determine the equation of l 3 in the form = m + b. Eperiential Activit Four Answers. + 6 = 30. = 0 3. = 4 Module A46 Linear Equation Theor -
OBJECTIVE FIVE When ou complete this objective ou will be able to Determine the slope, intercept, intercept, and/or the equation of a horizontal or vertical line. Eploration Activit HORIZONTAL LINE (l ) B definition an two points on a horizontal line have the same second coordinate. i.e. the same coordinate. (on l l all 's are 3) 6 l 4 (-, 3) (, 3) (, 3) l = 3-6 -4-4 6 - = -4 (-, ) -6 The SLOPE of a horizontal line ALWAYS = 0. NOTE: l and l refer to the previous figure. intercept of l. l does not cross - ais so it does not have an intercept intercept of l l crosses ais at = 3 so intercept is 3. Equation of l All points on l have a value of 3 so the equation of l is = 3. ( can be an real number). Module A46 Linear Equation Theor -
VERTICAL LINE(l ) B definition an two points on a vertical line have the same first coordinates (on l all 's are ). Slope of l m = 3 ( ) = ( ) 8 = 0 = undefined The slope of a vertical line is alwas undefined. intercept of l l crosses the ais at = so the intercept is. intercept of l l does not cross the ais so it does not have a intercept. Equation of l All points on l have an -value of so the equation of l is =. ( can be an real number). Important: Summar of Objective. Ever horizontal line has a slope of zero.. Ever vertical line has an undefined slope. 3. Horizontal lines do not have intercepts. 4. Vertical lines do not have intercepts.. Ever horizontal line has an equation of the form = k, where k is the intercept. For eample, = 3 is the equation of a horizontal line which crosses the ais at 3. 6. Ever vertical line has an equation of the form = k, where k is the intercept. For eample, = 6 is the equation of a vertical line which crosses the ais at 6. 6 Module A46 Linear Equation Theor -
Eperiential Activit Five l 6 4 l -6-4 - 4 6 - -4-6 From the previous figure:. Determine the following for l a. slope b. intercept c. intercept d. equation in standard form. Determine the following for l a. slope b. intercept c. intercept d. equation in standard form 3. Determine the equation of the line passing through the points (6, 7) and (6,4). Show Me. 4. Determine the equation of the line passing through the points (3, ) and ( 4, ). Show Me.. Determine the equation of the line passing through the points ( 4, 0) and ( 4,). Module A46 Linear Equation Theor - 7
Eperiential Activit Five Answers. a) 0 b) none c) d) =. a) undefined b) c) none d) = 3. = 6 4. =. = 4 8 Module A46 Linear Equation Theor -
OBJECTIVE SIX When ou complete this objective ou will be able to Appl the relationships that occur between lines that are parallel and those that occur between lines that are perpendicular. Eploration Activit Parallel lines: Parallel lines are two lines that etend in the same direction but do not intersect. Parallel lines have the same slope. Perpendicular lines: Perpendicular lines are two lines that intersect at 90. The slopes of perpendicular lines are the negative reciprocals of each other. (product of the two slopes = ) EXAMPLE Given the slope of l = 3, and l is parallel to l, and l is perpendicular to l 3. Find the slopes of l and l 3 SOLUTION:. Slope l = slope l parallel lines, slopes are equal. Therefore slope l = 3. Slope l = negative reciprocal l 3 (perpendicular lines) Therefore slope l 3 = 3 Module A46 Linear Equation Theor - 9
EXAMPLE Given line l with equation + = 7, find the slopes of l and l 3 if l is parallel to l and l 3 is perpendicular to l. SOLUTION: + = 7 Change the equation to the form = m + b so we can identif the slope, m. = 7 7 = Since the coefficient of is the slope we have: Slope l = Slope l = (l is parallel to l ) Slope l 3 = (l is perpendicular to l 3 ) 0 Module A46 Linear Equation Theor -
Eperiential Activit Si. Determine the equation, in the form A + B = C, of the line through (8, 7) and parallel to the -ais.. Determine the equation, in the form A + B = C, of the line through ( 3,) and perpendicular to the -ais. 3. Determine the equation, in the form = m + b, of the line through the origin and parallel to the line + + 3 = 0. 4. Determine the equation, in the form A + B = C, of the line having the same intercept as the line = and parallel to 3 =.. Determine the equation, in the form A + B = C, of the line containing ( 4, 3) and parallel to =. 6. Determine the equation, in the form = m + b, of the line through (, ) and perpendicular to + = 0. Show Me. 7. Determine the equation, in the form A + B = C, of the line having the same intercept as the line 3 = and parallel to 8 = 3. 8. Determine the equation, in the form A + B = C, of the line through ( 3, 4) and parallel to the ais. 9. Determine the equation, in the standard form, of the line through ( 7, 3) and parallel to the ais.. Determine the equation, in the standard form, of the line through ( 8, ) and perpendicular to the ais.. Determine the equation, in the standard form, of the line through (, ) and perpendicular to the ais.. Determine the equation, in the standard form, of the line through ( 3, 7) and perpendicular to 4 = 7. 3. Determine the equation, in the standard form, of the line through (, ) and parallel to 7 4 = 3. 4. Determine the equation, in the form = m + b, of the line with the same intercept as 3 + 4 = that is perpendicular to 3 = 7.. Determine the equation, in standard form, of the line with the same intercept as 3 = 9 that is perpendicular to 4 + 7 = 3. Module A46 Linear Equation Theor -
Eperiential Activit Si Answers. = 7.. = 3. = 4. 3 =. = 3 6. = + 7. 6 = 8. = 4 9. = 7. = 8. =. + 4 = 3 3. = 7 + 3 4 4 4. = 3 + 3. 4 8 = 63 Module A46 Linear Equation Theor -
OBJECTIVE SEVEN When ou complete this objective ou will be able to Solve word problems relating to straight lines. Eploration Activit EXAMPLE. Determine the perimeter of a triangle with vertices A(, 4), B(, ), C(, ). SOLUTION: 6 4 A(, 4) -6-4 - B(, ) 4 6 - C(, -) -4-6 Find the distance AB using the distance formula: d = ( ) + ( ) l =, = 4, =, = d(a, B) = ( ) + ( 4) = Module A46 Linear Equation Theor - 3
Find the distance BC using the distance formula: d = ( ) + ( ) =, =, =, = d(b, C) = ( ) + ( ) = Find distance AC, it's a vertical line therefore the distance = d = ( ) + ( ) =, = 4, = l, = d( A, C ) = ( ) + ( 4) = 6 Therefore the perimeter of triangle ABC = d(a, B) + d(b, C) + d(a, C) = + + 6 = 6 units 4 Module A46 Linear Equation Theor -
Eperiential Activit Seven. A triangle has vertices A( 6, ), B(4, ) and C(, 3). Determine the length of the median from C. (A median joins a verte to the midpoint of the opposite side).. A line segment has a midpoint at (, 3) and one endpoint at (, 8). Determine the other endpoint of the line. 3. The line through A(, ) and B( 3, 4) is parallel to the line through C(, 4) and D(, ). Determine the value of. Show Me. 4. The points T(, ), Q (4, 3), and R(, ) are on the same line (the are collinear). Determine the value of.. A right triangle contains one right angle. Given the vertices A(, ), B(, ) and C(3, 6). Find so that side AB is perpendicular to side BC. 6. The center of a circle is located at ( 3, ) and one of the endpoints of the diameter is at (, 7). Find the other endpoint of the circle s diameter. 7. A line passes through the points (, 8) and ( 7, 9). A line parallel to it passes through (3, ) and (8, ). Find the value of. 8. The points F(3, 9), G(7, ) and H(, ) are on the same line. Find the value of. 9. The altitude of a triangle is drawn from a verte of the triangle perpendicular to the opposite side of the triangle. If A( 3, 8), B(4, 7) and C(, ) are the vertices of a triangle, find the equation of the line that is the altitude drawn for B to side AC.. The coordinates of a rectangle are given b A(6, ), B( 3, ), C(, 6) and D(, 3). Find the length of a diagonal of the rectangle. Eperiential Activit Seven Answers.. (, ) 3. = 4. = 3. =3 6. ( 8, 3) 7. = 3.7 8. = 7 9. 9 = 9. 9.4868 Module A46 Linear Equation Theor -
Practical Application Activit Complete the Linear Equation - module assignment in TLM. Summar The topics presented in this Module dealt primaril with the algebra of equations of straight lines. B that we mean we studied:. Given the information about the line: find its equation.. Given the equation of a line: find information about it. Straight-line theor flows over into man varieties of applied problems. 6 Module A46 Linear Equation Theor -
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