LinearLinear EquationsEquations and Lines A linear equation in two variables is an equation that’s equivalent to an equationA linearof equationthe form in two variables is an equation that’s equivalent to an equation of the form ax+by+c=O ax + by + c = 0 where a, b, and c arjnstant numbers, and where a and b don’t both equal where0. (If the71otha, b, and c 41JIare constant0 then numbers,the polynomial and whereon thea andleftb sidedon’tof boththe equal equal 0.sign (Ifwouldn’t they werebe bothlinear.) 0 then the polynomial on the left side of the equal sign wouldn’t be linear.) Examples. Examples.

• 3x + 2y — 7 = 0 is a linear equation. • 3x + 2y − 7 = 0 is a linear equation.

• 2x = —4y +3 is a linear equation. It’s equivalent to 2x + 4y —3 = 0, • 2x = −4y + 3 is a linear equation. It’s equivalent to 2x + 4y − 3 = 0, and 2x + 4y — 3 is a linear polynomial. and 2x + 4y − 3 is a linear polynomial. • y = 2x +5 is a linear equation. It’s equivalent to the linear equation •y—2x5=0.y = 2x + 5 is a linear equation. It’s equivalent to the linear equation y − 2x − 5 = 0.

• x is a linear equation. = = y It’s equivalent to x — y 0. • x = y is a linear equation. It’s equivalent to x − y = 0.

• x = 3 is a linear equation. It’s equivalent to x — 3 = 0. • x = 3 is a linear equation. It’s equivalent to x − 3 = 0. • y = —2 is a linear equation. It’s equivalent to y + 2 = 0. • y = −2 is a linear equation. It’s equivalent to y + 2 = 0. IIaso4aInear equations are called linear Xbecause their solutions form%straightLinear equationsline5in are calledthe plane. linearWe’ll becausehave theirmore setsto ofsay solutionsabout these formlines straightin linesthe rest in theof this plane.chapter. We’ll have more to say about these lines in the rest of this chapter.

134 140 VerticalVertical lineslines The solutions of a linear equation x = c, where c ∈ is a constant, form a TheThe solutionssolutions ofof aa linearlinear equationequation xx == c,c, wherewhere cc EE RRRisis aa constant,constant, formform aa verticalvertical line:line: thethe thelineline lineofof ofallall allpointspoints pointsinin inthethe theplaneplane planewhosewhose whosex-coordinatesx-coordinatesx-coordinatesequalequal equalc.c.c.

CC

HorizontalHorizontalHorizontallineslines lines IfIfIfcccisisisaa aconstantconstant constantnumber,number, number,thenthen thenthethe thehorizontalhorizontal horizontallineline lineofof ofallall allpointspoints pointsinin inthethe theplaneplane plane whose y-coordinates equal c is the set of solutions of the equation y = c. whosewhose y-coordinatesy-coordinates equalequal cc isis thethe setset ofof solutionssolutions ofof thethe equationequation yy == c.c.

SlopeSlopeSlope The slope of a line is the ratio of the change in the second coordinate to the TheThe slopeslope ofof aa lineline isis thethe ratioratio ofofthethe changechange inin thethe secondsecond coordinatecoordinate toto thethe change in the first coordinate. In different words, if a line contains the two changechange inin thethe firstfirst coordinate.coordinate. InIn differentdifferent words,words, ifif aa lineline containscontains thethe twotwo points (x1, y1) and (x2, y2), then the slope is the change in the y-coordinate pointspoints (xi,(xi,y’)y’) andand (x2,(x2, y2),y2), thenthen thethe slopeslope isis thethe changechange inin thethe y-coordinatey-coordinate – which equals y − y – divided by the change in the x-coordinate – which — 2 1 — — — — which — which equals — divided by the change in the x-coordinate equals Y2Y2 — divided by the change in the x-coordinate whichwhich equals x2 − x1. — equalsequals X2X2 — x1.x1.

Slope of line containing (x , y ) and (x , y ): SlopeSlope ofof lineline containingcontaining (x1,(x1,1Yl)Yl)1 andand (x2,(x2,2y2):y2):2

y2 − y1 — Y2Y2 — YiYi x2 − x1 x2 — x2 —

141 135135 21 21 s)o: s)o:

—ç Example: The slope of the line containing the two points (−1, 4) and (2, −5) Example: The slope of the line containing the two points (—1,4) and (21—ç equals equals Example: The slope of the line containing the−two5 −points4 −(—1,4)9 and (21 —5—4 = = −3 equals 2 − (−1) 3 2—(—1)3—5—4 21 21I 2—(—1)3 s)o: po..ca1leI )LaXi po..ca1leIiies sope:—3 iies X a sope:—3

—ç Example: The slope of the line containingExample:the twoThepointsslope(—1,4)of theandline(21containing the two points (—1,4) and (2 equals equals Lines with the sameLinesslope withare theeither sameequal slopeor areparallel, either—5—4meaning equal orthey parallel,never meaning they never 5—4 intersect. intersect. 2—(—1)3 2—(—1) 3 Lines with the same slope are either equal or parallel, meaning they never intersect. po..ca1leI nvpraUeI iies hves sope:—3 s)ope:—3

142 Lines with 136the same slope are either equalLinesor withparallel,the meaningsame slopetheyarenevereither equal or parallel, meaning they never intersect. 136 intersect.

136 136 zii XaX,

Slope-intercept form for linear equations ZL a I In calculus, the most common form of linear equation you’ll see is y = ax+b, where a and b are constants. For example y = 2x − 5 or y = −2x + 7. An equation of the form y = ax + b is linear, because it’s equivalent to y − ax − b = 0. An equationExample: of the formThey slope= ax +ofbtheis calledline containing a linear equationthe two points (—1,4) and (2j) equals in slope-intercept form. 5—4 Claim: The solutions of the equation y = ax + b (where2—(—1)3a and b are numbers) form a line of slope a that contains the point (0, b) on the y-axis.

slope: a slope:—3

Proof: That (0, b) is a solution of y = ax + b is easy to check. Just replace Lines with the same slope are either equal or parallel, meaning they never x with 0 and y with b to see that intersect. (b) = b = a(0) + b The point (1, a + b) is also on the line for y = ax + b: (a + b) = a + b = a(1) + b Now that we know two points on the line, (0, b) and (1, a + b), we can find the slope of the line. The slope is (a + b) − b a = = a 1 − 0 1  S 136 Or —2(x)

P

siopt’ 2 Slope 143 Claim: y = ax is a line of slope a that contains the point (0,0). Claim: y = ax is a line of slope a that contains the point (0, 0).

Proof: This claim follows from the previous claim if we write y = ax as y Proof:= ax + 0.This Theclaim previousfollows claimfrom tellsthe usprevious that y =claimax + 0if awe linewrite of slopey = aaxthatas ycontains= ax +0. theThe pointprevious (0, 0). claim tells us that y = ax +0 a line of slope a that contains Point-slopethe point form(0,0). for linear equations Claim:AnotherLet commonL C R2 waybe the to writeline containing linear equationsthe point is as(p, (q)y −e qR2) =anda(havingx − p), p).’S slope equal to the number a. Then t eqtn$-Eri - = where a, p, q ∈ R are constants. This is called the point-slope(y formq) ofa(x a— linear equation.Before writing Examplesthe includeproof for (y −this2) =claim, 7(x −let’s3) andlook (yat− an1) =example.−2(x + 4).Let’s suppose that we2r4t2€re Zi1ing an equation for a line 2whose slope Claim: Let L ⊆ R be the line containing the point (p, q) ∈ R and having isslope—2, equaland that to thepasses numbert1,irougha. Thenthe point (y − qin) =thea(planex − p)(4, is1). an equationThen4he forclaimL. says that we can use LTf equation i) = 2(x — 4). (y — We may prefer to As an example of the claim, (y − 1) = −2(x − 4) is a line of slope −2 that simplify this equation as y — 1 = 2x — 8 or y = 2x — 7. passes through the point in the plane138 (4, 1). 144 ______

(z 2) 211 ltai S S Or —2(x)Or —2(x) XaXI

P P L_d Xl xa —.X, siopt’ 2siopt’ 2 Slope Slope *5; Example: The slope of the line containing the twoExample:points (—1,4)The slopeand (2()of the line containing the two points (—1,4) and (2j Now let’sNow turnlet’s toreturn the proofto the ofproof the claim. equals of the claim.equals Proof: Let S be the set of solutions of y = ax. As shown in the previous 5—4 - Proof: Let S be the set54of solutions of y = ax. As shown in the previous ; claim, claim,S is aS lineis a inline thein planethe2-(-1)3plane whosewhose slope equalsslope equalsa anda thatand containsthat contains the the2-(-1)3 point (0point, 0). (0,0). S . (‘..l, (p.) Slope s)ope:—3 slope:—3 (0,0) (a, a)

r (2,-5)

We canWe usecan theuse additionthe addition functionfunctionA(p,q) toA(p,q) shiftto theshift linetheS linehorizontallyS horizontally by by Lines with the same slope are either equal or parallel, p,Lines andp, verticallywithandtheverticallysame by q.slope Thisby q. newareThiseither line,newAequalline,(p,q)(SA(p,q)(S),or),parallel, is parallelis parallelmeaning to ourto originaltheyour neveroriginal line, line, meaning they never intersect. intersect.so its slopeso its alsoslope equalsalso equalsa. Thea. lineTheA(linep,q)(SA(pq)(S)) also containsalso contains the pointthe point (p, q), (p, q), becausebecause (0, 0) ∈(0,0)S andS thusand (p,thus q) =(p,Aq)(p,q=)(0A(p,q)(0,, 0) ∈ A(0))p,q)(S).A(p,q)(S). BecauseBecauseA(p,q)(SA(p,q)) has(5) slopehas aslopeanda containsand contains the pointthe point (p, q),(p, weq), seewe thatsee that A(p,q)(SA(p,q)) is the(5) lineis theL fromline L thefrom claimthe whoseclaim whose equationequation we wouldwe would like tolike identify,to identify, and weand canwe findcan thefind equationthe equation forA(p,qforA(p,q)(S))(S) = L using= L POTS:using POTS: ojorS

—A 1 (pry) I I.

16 136 (-‘a(xp)  145 139 Problem: Find an anequation equationfor forthe theline linecontaining containingthe thetwo twopoints points(1,3) (1, 3) and ((—2,−2, −—4)4).

3)

Solution: First we’ll we’llfind findthe theslope slopeof ofthe theline. line.It’s It’s —4—−4 − 33 —7−7 77 = =

—2−2 —− 1 — —3−3 — 33 Second, we’ll use usethe theprevious previousclaim claimwhich whichtells tellsus usthat thata aline linecontaining containingthe the 7 point (1,3) (1, 3)and with withslope slope 3 hashasas asan anequation equation 7 (y-3)=(x-1)(y − 3) = (x − 1) 3

140146 Exercises

For #1-4, find the slope of the line that contains the two given points in the plane.

1.) (1, 2) and (3, 4) 3.) ( 2, 3) and (2, 1) 2.) (5, 10) and (0, 0) 4.) (3, 2) and ( 6, 7)

For #5-10, identify the slope of the line that is the set of solutions of the given linear equation in slope-intercept form.

5.) y =2x +3 8.) y = x +5

6.) y = 4x +7 9.) y =17 7.) y = x 210.)y =3x 4

For #11-16, identify the number b with the property that (0,b)isapoint in the line of the solutions of the given linear equation. This number is called the y-intercept.

11.) y =3x +8 14.) y =9

12.) y =2x +9 15.) y =5x 8 13.) y = 3x 716.)y =2x +5

For #17-20, write an equation for a line that has the given slope and contains the given point. Write answers in point-slope form (y q)=a(x p).

17.) Slope: 4. Point: (2, 3). 19.) Slope: 1. Point: (1, 5). 18.) Slope: 2. Point: ( 3, 4). 20.) Slope: 7. Point: ( 4, 3). 147 For #21-24, provide an equation for a line that contains the given pair of points. Write your answers in the slope-intercept form y = ax + b.

21) (8, 2) and (2, 5) 23.) ( 3, 5) and (3, 4) 22.) (7, 3) and (2, 1) 24.) (2, 8) and ( 2, 5)

Find the inverse matrices.

1 1 32 4 3 25.) 26.) 1 2 21 ✓ ◆ ✓ ◆

Find the solutions of the following equations in one variable.

2x 3 2 27.) e =0 28.)log3(x) +3=4log3(x)

2 Recall that pX : R R where pX(x, y)=x is called the projection onto ! 2 the x-axis, and that pY : R R where pY (x, y)=y is the projection onto the y-axis. In the remaining questions,! find the appropriate values.

29.) pX(3, 7) 33.) pX(1, 8)

30.) pX 2, 8 34.) pY 6, 9

31.) p ( 2, 5) 35.) p ( 1,7) Y Y

32.) pY (4, 0) 36.) pX(2, 3)

148