2b:null 2c:["$","title",null,{"children":"Linear Equations & Parameters | Murat Anlı | Observable"}] 31:[] 2f:{"id":"9dfd80b8bf54f183","slug":null,"title":"Linear Equations & Parameters","thumbnail":"8744fdd17681c40b5d6bc1906dd2e55efb2cc3c6aa1be3149191349f401d022c","access_level":"public","create_time":"$D2026-03-29T19:42:42.455Z","update_time":"$D2026-03-29T19:45:00.255Z","view_time":null,"first_public_time":"$D2026-03-29T19:42:42.455Z","has_unpublished_edits":null,"fork":{"id":"20f4ff15a117bde3","slug":null,"title":"Linear Equations & Parameters","owner":{"id":"099e5d662aa22e29","login":"omanli","name":"Murat Anlı"}},"stars":0,"starred":false,"owner":{"type":"team","id":"099e5d662aa22e29","login":"omanli","name":"Murat Anlı","avatar_url":"https://avatars-next.observableusercontent.com/a4d53bc536a70a0abc8fc18b8c0b1f04c29d7ac0e7d88d15f9041ce9fd58dbe9","tier":"starter_2024"},"version":1374,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"58c0ca80f1801961","login":"omanli","name":"Murat Anlı","avatar_url":"https://avatars-next.observableusercontent.com/a4d53bc536a70a0abc8fc18b8c0b1f04c29d7ac0e7d88d15f9041ce9fd58dbe9","tier":null}],"members":[{"type":"individual","id":"58c0ca80f1801961","login":"omanli","name":"Murat Anlı","avatar_url":"https://avatars-next.observableusercontent.com/a4d53bc536a70a0abc8fc18b8c0b1f04c29d7ac0e7d88d15f9041ce9fd58dbe9","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":0,"collection":null,"type":"notebook","body":{"title":"Linear Equations & Parameters","cells":[{"id":1166,"value":"# Linear Equations & Parameters\n## MATH302 Numerical Analysis","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1338,"value":"An Equation vs a Function\n\n${tex`\\quad \n\\begin{aligned}\nx^3 - x^2 & = e^{-4x+5} \\\\[2ex]\nf(x) & = \\dfrac{x^2 - 3x + 10}{x - 4}\n\\end{aligned}\n`}\n\nEquals sign (${tex`=`}) implies two very different things. Equals sign in an equation implies a problem or a puzzle to be solved. Equals sign in a function implies a definition or a rule.\n\n${tex`\\quad \n\\begin{aligned}\n(x - x_0)^2 + (y - y_0)^2 & = r^2 \\\\[2ex]\ng(u, v, w) & = 3uv - 5v(u + w) - 2uvw\n\\end{aligned}\n`}\n\n\n","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":13,"value":"Let's dive into some basic uses of equations and functions. We will explore alternative ways to express _lines_ in two dimensions as linear equations and how those expressions will yield different conveniences and help communicate different points of view.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":7,"value":"### Linear Equations","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":30,"value":"A linear equation is one that can be expressed as (or converted into) linear algebraic calculations. A linear equation expresses or represents a _line_. This line can have 2 or more dimensions.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":40,"value":"#### Example:\n\nLet's start with the following equation: ${tex`\\quad x + y = 10`}\n\nThis equation has two variables (unknowns) and it is linear. As linear equation it describes all the points ${tex`(x,y)`} that belong to a straight line. This equation defines membership to this line. If point ${tex`(x,y)`} satisfies the equation then this point is a member of the line and otherwise it is not part of the line. For example ${tex`(10,0)`}, ${tex`(0,10)`}, ${tex`(5,5)`}, ${tex`(7,3)`}, ${tex`(-4,14)`}, ${tex`(320,-310)`} are all on this line (i.e. members of this line). We can view a line as a set and the points on this line are members of this set. For example none of ${tex`(1,1)`}, ${tex`(34,52)`}, ${tex`(9,2)`} are this line.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1370,"value":"

","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1196,"value":"Plot.plot({\n grid: true,\n x: { domain: [x_min, x_max], ticks:x_ticks },\n y: { domain: [y_min, y_max], ticks:y_ticks },\n width: plot_width,\n height: plot_height,\n marks: [\n Plot.ruleX([0], { strokeWidth: axes_width }),\n Plot.ruleY([0], { strokeWidth: axes_width }),\n Plot.axisY({x:0, insetLeft: 3, textStroke: \"white\", fontSize:tick_fs, ticks:y_ticks}),\n Plot.axisX({y:0, insetBottom: 3, textStroke: \"white\", fontSize:tick_fs, ticks:x_ticks}),\n Plot.line(PLine1, {x: \"x\", y: \"y\", z:\"id\", stroke:\"id\", r: \"id\", strokeWidth:line_width }),\n //Plot.dot(PDot, {x:\"x\", y:\"y\", z:\"id\", stroke:\"id\", fill:\"id\", r:dot_size}),\n ],\n color: {\n domain: Array.from(new Array(20),(val,i)=> String(i+1)),\n legend: false\n }\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":77,"value":"Working with linear equations the first form we will look at is:\n${tex.block`ax + by = c`}\nThe coefficients in the equation can be labeled via different conventions. Some of the alternatives:\n${tex.block`\n \\begin{array}{rcrcrcr}\n ax &+& b y &+& c & = & 0 \\\\[2ex]\n c_1 x &+& c_2 y &+& c_3 & = & 0 \\\\[2ex]\n c_x x &+& c_y y &+& c & = & 0 \\\\\n \\end{array}\n`}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":115,"value":"Some of the interesting properties of a line are its: \n-- slope \n-- x-intercept \n-- y-intercept \nCan we readily find these properties of a given line if it is given in the first expression style as above?","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":132,"value":"For example examine the following equation: \n${tex.block`\\text{Line Equation:}\\quad ax + by = c`}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":341,"value":"We need to calculate each of the aforementioned properties: \n${tex.block`\n \\begin{aligned}\n \\text{slope} & = -\\frac{\\,a\\,}{b} \\\\[3ex]\n \\text{x-intercept} & = \\frac{\\,c\\,}{a} \\\\[3ex] \n \\text{y-intercept} & = \\frac{\\,c\\,}{b}\n \\end{aligned}\n`}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1266,"value":"define \n${tex.block` m,s,x_0,y_0,c,y_1,x_1`}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":182,"value":"Would it be easier if the line equation were expressed as:\n${tex.block` y = ax + b `}\nEquivalently we can different labels for the constants:\n${tex.block`\n \\begin{array}{rcrcr}\n y & = & mx & + & b \\\\[2ex]\n y & = & mx & + & c \\\\[2ex]\n y & = & sx & + & c\n \\end{array}\n`}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":189,"value":"Notice that labels do not change the nature of the equation:\n${tex.block`\n\\begin{array}{rcrcr}\ny & = & \\text{}x & + & \\text{} \\\\[2ex]\ny & = & \\text{}x & + & \\text{}\n\\end{array}\n`}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":193,"value":"And similarly:\n${tex.block`\\text{}x + \\text{}y = \\text{}`}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":201,"value":"Furthermore manipulations of the equation expression does not change the nature of the equation either:\n${tex.block`\\text{}x + \\text{}y + \\text{} = 0`}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":222,"value":"#### Example: \nAll of the following equations are equivalent: \n${tex.block`\\begin{array}{rcrcrcr}\n 4x & + & 3y & - & 9 & = & 0 \\\\[1.5ex]\n & & 4x & + & 3y & = & 9 \\\\[1.5ex]\n & & 40x & + & 30y & = & 90 \\\\[1.5ex]\n & & \\llap{$-$}20x & - & 15y & = & -45 \\\\[1.5ex]\n-20x & - & 15y & + & 45 & = & 0 \\\\[3ex]\n\\end{array}`}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":797,"value":"The group of equations above have the variables ${tex`x`} and ${tex`y`} on the left-hand-side. First and the last equations have a zero right-hand-side. The group of equations below have one of the variables on the left-hand-side and everything else on the right-hand-side. \n${tex.block`\n\\begin{array}{rcrcr}\n -3y & \\llap{=} & ~~~~~~~\\llap{$4x$}~ & \\llap{$-$} & ~~\\llap{9\\,\\,} \\\\[2.5ex]\n 3y & \\llap{=} & \\llap{$-4x$}~ & \\llap{$+$} & ~~\\llap{9\\,\\,} \\\\[2.5ex]\n y & \\llap{=} & \\llap{$-\\dfrac{\\,4\\,}{3}x$}~ & \\llap{$+$} & ~~\\llap{3\\,\\,} \\\\[2.5ex]\n 4x & \\llap{=} & \\llap{$-3y$}~ & \\llap{$+$} & ~~\\llap{9\\,\\,} \\\\[2.5ex]\n x & \\llap{=} & \\llap{$-\\dfrac{\\,3\\,}{4}y$}~ & \\llap{$+$} & \\llap{$\\dfrac{\\,9\\,}{4}$}\n\\end{array}\n`}\n","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":317,"value":"We can get the properties we are interested in by either calculating it or from the appropriate expression form:\n${tex.block`\\begin{aligned}\n\\text{slope} & = -\\dfrac{\\,4\\,}{3} \\\\[2ex]\n\\text{x-intercept} & = \\dfrac{\\,9\\,}{4} \\\\[2ex]\n\\text{y-intercept} & = \\,\\,3\n\\end{aligned}\n`}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":205,"value":"What does it mean to be *same equation*? \nWhat does it mean for two equations to be *equivalent*? \nIf two equations are *equivalent* then all values (points) that satisfy the first equation must satisfy the second one and all values that do not satisfy the first equation must not satisfy the second one. This basically says that the set of points defined by the first line equals the set of points of defined by the second line.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":250,"value":"Let's look at the following form:\n${tex.block`\n\\begin{aligned}\n(y - y_1) & = m(x - x_1) \\\\[2ex]\n(y - \\text{}) & = \\text{}(x - \\text{})\n\\end{aligned}\n`}\nThis form is the answer to the question: \n\"What is the equation of the line that \n -- passes through point ${tex`(x_1,y_1)`}, and \n -- has slope ${tex`m`}?\" \nFurthermore we can convert equations from one form to another and easily calculate the quantities that we are interested in.\n${tex.block`\n\\begin{aligned}\n\\text{y-intercept} = y_0 & = y_1 - mx_1 \\\\[2ex]\n\\text{x-intercept} = x_0 & = \\frac{mx_1 - y_1}{m} = x_1 - \\frac{y_1}{m} \n\\end{aligned}\n`}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":304,"value":"#### Example:\nLet's convert\n${tex.block`y = 5x - 2`}\ninto the slope&point form. \nNotice that the slope is 5. Now we need a point ${tex`(x,y)`} on this line. If do not prefer any specific point then any point on the line will do. All of these points are on this line: ${tex`(1,3)`}, ${tex`(4,18)`}, ${tex`(-2,-12)`}, ${tex`(3,13)`}, ${tex`(-1,-7)`} which means all the following equations are equivalent expressions for the line above:\n${tex.block`\\begin{aligned}\n(y - 3) & = 5(x - 1) \\\\[2ex]\n(y - 18) & = 5(x - 4) \\\\[2ex]\n(y + 12) & = 5(x + 2) \\\\[2ex]\n(y - 13) & = 5(x - 3) \\\\[2ex]\n(y + 7) & = 5(x + 1)\n\\end{aligned}`}\n","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1055,"value":"If two line equations with different expressions yield the same slope and the same y-intercept\ncan they be different lines? Equivalently, how many lines can have the same slope and y-intercept? If the slope and the y-intercept is given, is the line uniquely identified? The answer to all these questions is \"yes\". If we generalize it we could say any one of the following information uniquely identifies a line: \n--slope & y-intercept \n--slope & x-intercept \n--slope & a point ${tex`(x_0,y_0)`} on the line \n--${tex`(x_1,y_1)`} & ${tex`(x_2,y_2)`}, two different points on the line \n--x-intercept & y-intercept \nIn short what it takes to uniquely identify a line is either a point on the line and the slope of the line, or two different points on the line.\nReminder: the x-intercept and the y-intercept are basically examples of points on the line of the following form: ${tex`(x_0,0)`} & ${tex`(0,y_0)`}. \n\n","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1092,"value":"$${tex`m`} & ${tex`y_0`}, slope and y-intercept:\n${tex.block`y = mx + y_0`}\n\n${tex`m`} & ${tex`x_0`}, slope and x-intercept:\n${tex.block`x = \\dfrac{1}{\\,m\\,}y + x_0`}\n\n${tex`m`} & ${tex`(x_1,y_1)`}, slope and a point on the line:\n${tex.block`(y - y_1) = m(x - x_1)`}\n\n${tex`(x_1,y_1)`} & ${tex`(x_2,y_2)`}, two different points on the line:\n${tex.block`\\dfrac{\\,y - y_1\\,}{\\,x - x_1\\,} = \\dfrac{\\,y - y_2\\,}{\\,x - x_2\\,}`}\n\n${tex`x_0`} & ${tex`y_0`}, x-intercept and y-intercept:\n${tex.block`\\dfrac{\\,x\\,}{\\,x_0\\,} + \\dfrac{\\,y\\,}{\\,y_0\\,} = 1`}\n\n\nNote: the slope can be calculated as follows in the last two forms. \nIf two different points on the line, ${tex`(x_1,y_1)`} & ${tex`(x_2,y_2)`}, are given:\n${tex.block`\\text{slope} = \\dfrac{\\,y_2 - y_1\\,}{\\,x_2 - x_1\\,} = \\dfrac{\\,y_1 - y_2\\,}{\\,x_1 - x_2\\,}`}\nIf x-intercept and y-intercept, ${tex`x_0`} & ${tex`y_0`}, are given:\n${tex.block`\\text{slope} = -\\dfrac{\\,y_0\\,}{\\,x_0\\,}`}\n","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1086,"value":"If we are given two different linear expressions, how do we check that they express the same line or two different lines? We just need to simplify the equations into the slope & intercept form ${tex`y = mx + y_0`}. If both of them have the same slope and same y-intercept then the lines defined by these two equations are the same line. Otherwise we have two different lines.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1142,"value":"Does every line have an x-intercept and a y-intercept? \nIf the slope is not ${tex`0`} or ${tex`\\infty`} then that line has finite valued x-intercept and y-intercept. Otherwise the line is either parallel to the x-axis or the y-axis, which means that it does intersect one of the axes and hence has only one intercept not two. ","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1312,"value":"

","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":58,"value":"data = [\n { ex: 1, point:1, x: -5, y: 15 },\n { ex: 1, point:2, x: 0, y: 10 },\n { ex: 1, point:3, x: 10, y: 0 },\n { ex: 1, point:4, x: 20, y: -10 }\n ]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":669,"value":"functionPlot = require(\"https://unpkg.com/function-plot@1/dist/function-plot.js\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1200,"value":"plot_width = 600","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1206,"value":"plot_height = 400","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1209,"value":"x_min = -10","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1211,"value":"x_max = 20","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1213,"value":"y_min = -10","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1217,"value":"y_max = 20","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1219,"value":"x_ticks = 10","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1221,"value":"y_ticks = 10","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1223,"value":"tick_fs = 14","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1240,"value":"line_width = 4","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1225,"value":"axes_width = 3","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1238,"value":"PLine1 = [{x:\"-110\", y:\" 120\", id:\"1\"},\n {x:\" 210\", y:\"-200\", id:\"1\"}]","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1284,"value":"$$css = html`\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1282,"value":"$$ = {\n $css;\n var $ = tex.options({trust: true});\n $.options = (opts) => tex.options(Object.assign({trust: true}, opts));\n return $;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1286,"value":"$$$ = $css, tex.options({displayMode: true, trust: true})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W31","stdlib":"1","titling":"auto"}} 30:"$2f:body"