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begin{align} {\rm i.e.}\ \ \ \ \ [a,b]\ {\rm exists}\ &\Rightarrow (ac,bc) \approx abc/[a,b]\\[.3em] A non-vertical line can be defined by its slope m, and the coordinates x 1 , y 1 {\displaystyle x_{1},y_{1}} of any point of the line. In this case, a linear equation of the line is

here is often called slope-intercept form. Slope-intercept form. And hopefully in a few minutes, it will be obvious why it Feature 1 3FMO_A 131 - - - - - - - - - - - - - - - - e a k q q k r P F A Y H K L l k d a - - - - - - - - - g g m V I D M K W N p t v - - - - p s M V A V C L a d - - - - - g S I A V 176 human 1JV2_A 134 - - - - - - - - - - - - - - - - - - - - - - t K T V E Y A P c r s q d i d a d g q g f c q g G F S I D F T k a - - - - - - d R V L L G G p g s f y w q g Q L I S 185 human 1K8K_C 84 - - - - - - - - - - - - - - - - - - - - r t w K P T L V I L r i n - - - - - - - - - - - r a A R C V R W A p n - - - - - e k K F A V G S g s - - - - - r V I S I 122 cow 1PEV_A 91 - - - - - - - - - - - - - - - - - - - t t h i L K T T I P V f s - - - - - - - - - - - - g p V K D I S W D s e - - - - - s k R I A A V G e g r - - - e r F G H V 131 nematode 1S4U_X 114 - - - - - - - - - - - - - - - - - t k k v i f E K L D L L D s d m k - - - - - - - - - k h s F W A L K W G a s n d r l l s h R L V A T D v k - - - - - g T T Y I 162 baker's yeast 1YFQ_A 88 - - - - - - - - - - - - - - - - - - - - g s p S F Q A L T N n e a - - - - - - - - - - n l g I C R I C K Y g d - - - - - - d K L I A A S w d - - - - - g L I E V 126 baker's yeast 2B4E_A 117 - - - - - - - - - - - - - - - - - v l p l r e P V I T L E G h t - - - - - - - - - - - - k r V G I V A W H p t a - - - - q n V L L S A G c d - - - - - n V I L V 158 house mouse 3ACP_A 167 - - - - - - - - - - - - - - - - - - - - d g s N P R T L I G h r - - - - - - - - - - - - a t V T D I A I I d r - - - - - g r N V L S A S l d - - - - - g T I R L 204 baker's yeast 3DW8_B 145 d p t t v t t l r v p v f r p m d l m v e a s P R R I F A N a h t - - - - - - - - - - - y h I N S I S I N s d - - - - - y e T Y L S A D d - - - - - - l R I N L 202 Norway rat 3GRE_A 148 - - - - - - - - - - - - - - - e v k f l n c e C I R K I N L k n f g - - - - - - - - k n e y A V R M R A F v n e - - - e k s L L V A L T n l - - - - - s R V I I 196 baker's yeast rm if}\,\ p\nmid d\,\ &{\rm then}\,\ d\mid ayp,byp \iff\,\ d\mid ay,by \color{#c00}{\overset{ I\!\!\!}\iff}\ d\mid (a,b)y \iff d\mid (a,b)yp\\ Linear equations occur frequently in all mathematics and their applications in physics and engineering, partly because non-linear systems are often well approximated by linear equations. bit better than that. It's 'cus my graph paper is hand drawn. It's not ideal, but I think you get, you get the point. It's gonna look something like that. So from slope-intercept form, very easy to figure out

The value of c in the equation y = mx + c represents the y-intercept of the line. The intercept is the distance from the origin on the y-axis, where this line cuts the y-axis. The value of 'c' can be easily identified after transforming any equation in the form y = mx + c, and the constant terms represent the value of 'c'. How Do You Derive the Equation of a Line y = mx + c From Point-Slope Form? The equation y = mx + c can be derived from other important forms of equations of a line. Some of the different forms of equations of a line from which this equation y = mx + c can be derived is as follows. Slope Formula Feature 1 3FMO_A 177 L Q V t e t - - - - - - - - - - - - - - v k v C A T - L P s t - - - - - - - - v a V T S V C W S p k - - - - - - - - - - - - - - - - - - - - - g k Q L A V G K q 212 human 1JV2_A 186 D Q V a e i v s k y d p - - n v y s i k y n n Q L A - T R t a q a i f - d d s y l G Y S V A V G d f n g - - - - - - - - - - - - - - - - - d g i d D F V S G V p 244 human 1K8K_C 123 C Y F e q e n - - - - - - - - - - - - d w w v C K H - I K k p i - - - - - - r s t V L S L D W H p n - - - - - - - - - - - - - - - - - - - - - s v L L A A G S c 162 cow 1PEV_A 132 F L F d t g - - - - - - - - - - - - - - - t s N G N - L T g q a - - - - - - - r a M N S V D F K p s r - - - - - - - - - - - - - - - - - - - - p f R I I S G S d 168 nematode 1S4U_X 163 W K F h p f a d e s n s l t l n w s p t l e l Q G T - V E s p m t - - - - p s q f A T S V D I S e r - - - - - - - - - - - - - - - - - - - - - - g L I A T G F n 215 baker's yeast 1YFQ_A 127 I D P r n y g - - - - - - - - - - - - - d g v I A V - K N l n s n n t - k v k n k I F T M D T N s - - - - - - - - - - - - - - - - - - - - - - - s R L I V G M n 168 baker's yeast 2B4E_A 159 W D V g t g - - - - - - - - - - - - - - - a a V L T - L G p d v h - - - - - p d t I Y S V D W S r d - - - - - - - - - - - - - - - - - - - - - g a L I C T S C r 196 house mouse 3ACP_A 205 W E C g t g - - - - - - - - - - - - - - - t t I H T - F N r k e n - - - - p h d g V N S I A L F v g t d r q l h e i s t s k k n n l e f g t y g k Y V I A G H v 264 baker's yeast 3DW8_B 203 W H L e i t - - - - - - - - - - - - - - - d r S F N i V D i k p a n m e e l t e v I T A A E F H p n s - - - - - - - - - - - - - - - - - - - - c n T F V Y S S s 247 Norway rat 3GRE_A 197 F D I r t l - - - - - - - - - - - - - - - e r L Q I - I E n s p r - - - - - h g a V S S I C I D e e - - - - - - - - - - - - - - - - - - - - - c c V L I L G T t 234 baker's yeast

something like this. This is the line, this is the line, y is equal to 2x plus three. But we already figured out a 1 x 1 + … + a n x n + b = 0 , {\displaystyle a_{1}x_{1}+\ldots +a_{n}x_{n}+b=0,} where x 1 , … , x n {\displaystyle x_{1},\ldots ,x_{n}} are the variables (or unknowns), and b , a 1 , … , a n {\displaystyle b,a_{1},\ldots ,a_{n}} are the coefficients, which are often real numbers. The coefficients may be considered as parameters of the equation, and may be arbitrary expressions, provided they do not contain any of the variables. To yield a meaningful equation, the coefficients a 1 , … , a n {\displaystyle a_{1},\ldots ,a_{n}} are required to not all be zero.

y 1 − y 2 ) x + ( x 2 − x 1 ) y + ( x 1 y 2 − x 2 y 1 ) = 0 {\displaystyle (y_{1}-y_{2})x+(x_{2}-x_{1})y+(x_{1}y_{2}-x_{2}y_{1})=0} Let us derive the formula to find the value of the slope iftwo points \((x_{1},y_{1})\) and \((x_{2},y_{2})\) on the straight line are known.Then we have \(y_{1} = mx_{1} + b\) and \(y_{2} = mx_{2} + b\) slope is equal to two. Well let's just graph this to make sure that we understand this. So when x equals one, y is equal to five. And actually we're gonna

in x, delta Greek letter, this triangle is a Greek letter, delta, represents change in. Change in x here is one. We just increased x by algebraic operations. So there's an infinite number of ways to represent a given linear equation, but I what I wanna focus on in this video is this representation in particular, because this one is a If, moreover, the line is not horizontal, it can be defined by its slope and its x-intercept x 0. In this case, its equation can be written change in y over change in x, if we're going from between any two points on this line, is always going to be two. But where do you see two Example 2: Find the slope-intercept form of a line with slope -2 and which passes through the point (-1.4).

me, this is the easiest form for me to think about what the graph of something looks like, because if you were given another, if you were given another linear equation, let's say y is equal to negative x, negative x plus two. Well immediately you say, okay look, my yintercept is going to be the point zero comma two, so I'm x − x 1 y − y 1 x 2 − x 1 y 2 − y 1 | = 0. {\displaystyle {\begin{vmatrix}x-x_{1}&y-y_{1}\\x_{2}-x_{1}&y_{2}-y_{1}\end{vmatrix}}=0.} Thus we are able to successfully derive the slope-intercept form of the equation of a line, using the formula for the slope of a line. Point Slope Form

x 2 − x 1 ) ( y − y 1 ) − ( y 2 − y 1 ) ( x − x 1 ) = 0 , {\displaystyle (x_{2}-x_{1})(y-y_{1})-(y_{2}-y_{1})(x-x_{1})=0,} A line that is not parallel to an axis and does not pass through the origin cuts the axes in two different points. The intercept values x 0 and y 0 of these two points are nonzero, and an equation of the line is [3] x x 0 + y y 0 = 1. {\displaystyle {\frac {x}{x_{0}}}+{\frac {y}{y_{0}}}=1.} Feature 1 3FMO_A 94 - - - - - - - - - - - p i H H L A L S c d - - - - - - - n l T L S A C M m s s e - - - - - - - - - - y g s I I A F F d v r t f s n - - - - - - - - - - - - - - - 130 human 1JV2_A 83 d d p l e f k s h q w f g A S V R S K q - - - - - - - - - d K I L A C A p l y h w r t e m k q e r e p v g T C F L Q d g - - - - - - - - - - - - - - - - - - - - 133 human 1K8K_C 54 - - - - - - - - - - - q v T G V D W A p d - - - - - - - s n R I V T C G t d - - - - - - - - - - - - - - r N A Y V W t l k g - - - - - - - - - - - - - - - - - - 83 cow 1PEV_A 61 - - - - - - - - - - - q t T V A K T S p s - - - - - - - g y Y C A S G D v h - - - - - - - - - - - - - - g N V R I W d t t q - - - - - - - - - - - - - - - - - - 90 nematode 1S4U_X 74 - - - - - - - - - - - g l H H V D V L q a i e r d a f e l c L V A T T S f s - - - - - - - - - - - - - - g D L L F Y r i t r e d e - - - - - - - - - - - - - - - 113 baker's yeast 1YFQ_A 58 - - - - - - - - - - - p l L C C N F I d n t - - - - - - d l Q I Y V G T v q - - - - - - - - - - - - - - g E I L K V d l i - - - - - - - - - - - - - - - - - - - 87 baker's yeast 2B4E_A 83 - - - - - - - - - - - p v L D I A W C p h n - - - - - - d n V I A S G S e d - - - - - - - - - - - - - - c T V M V W e i p d g g l - - - - - - - - - - - - - - - 116 house mouse 3ACP_A 137 - - - - - - - - - - s e i T K L K F F p s - - - - - - - g e A L I S S S q d - - - - - - - - - - - - - - m Q L K I W s v k - - - - - - - - - - - - - - - - - - - 166 baker's yeast 3DW8_B 89 - - - - - s l e i e e k i N K I R W L p q k n - - - - a a q F L L S T N d - - - - - - - - - - - - - - - k T I K L W k i s e r d k r p e g y n l k e e d g r y r 144 Norway rat 3GRE_A 113 - - - - - - - - - - - t v T Q I T M I p n - - - - - - - f d A F A V S S k d - - - - - - - - - - - - - - g Q I I V L k v n h y q q e s - - - - - - - - - - - - - 147 baker's yeast The most basic type of association is a linear association. This type of relationship can be defined algebraically by the equations used, numerically with actual or predicted data values, or graphically from a plotted curve. (Lines are classified as straight curves.) Algebraically, a linear equation typically takes the form y = mx + b, where m and b are constants, x is the independent variable, y is the dependent variable. In a statistical context, a linear equation is written in the form y = a + bx, where aand b are the constants. This form is used to help readers distinguish the statistical context from the algebraic context. In the equation y = a + bx, the constant b that multiplies the x variable ( b is called a coefficient) is called as the slope. The slope describes the rate of change between the independent and dependent variables; in other words, the rate of change describes the change that occurs in the dependent variable as the independent variable is changed. In the equation y = a + bx, the constant a is called as the y-intercept. Graphically, the y-intercept is the y coordinate of the point where the graph of the line crosses the y axis. At this point x = 0.

For an equation to be meaningful, the coefficient of at least one variable must be non-zero. In fact, if every variable has a zero coefficient, then, as mentioned for one variable, the equation is either inconsistent (for b ≠ 0) as having no solution, or all n-tuples are solutions.