[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-605560-105":59,"doc-detail-605560-en":130},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","chapter-11-systems-of-equations-equation-or-function","Chapter 11 - Systems of Equations - Equation or Function?","","Chapter 11 explains the distinction between equations and functions and then introduces systems of equations as a set of two or more equations sharing the same variables. It walks through graphing lines from two linear equations by finding x- and y-intercepts, showing that intersecting lines reveal a common solution point. The chapter then motivates solving systems without graphing, presenting substitution as a method for two-variable systems and beginning elimination as an alternative approach.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":20,"@type":76,"position":81},"https://docshare.wps.com/document/exam/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/chapter-11-systems-of-equations-equation-or-function/605560/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/chapter-11-systems-of-equations-equation-or-function/605560.png","ImageObject",300,407,{"name":92,"@type":93},"WPS_1790064749","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-30","2026-10-16",true,{"@type":102,"interactionType":103,"userInteractionCount":29},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What is the difference between an equation and a function in this chapter?","Question",{"text":112,"@type":113},"An equation is treated as something to be solved for the unknown value (often x), without a graph expressing dependence. A function describes how an output value changes based on the input x, such as f(x)=x^3−x determining a height.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How do you find and graph a system of two equations?",{"text":117,"@type":113},"Graph each equation on the coordinate plane by locating intercepts. Substitute x=0 to get the y-intercept and substitute y=0 to get the x-intercept for each line, then use two points to draw each line.",{"name":119,"@type":110,"acceptedAnswer":120},"How does substitution solve a two-variable system?",{"text":121,"@type":113},"Solve one equation for one variable in terms of the other, then substitute that expression into the other equation. This reduces the system to a single-variable equation, after which you back-substitute to find the remaining variable.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},605560,1792461173,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":29,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":144,"read_time":145},3985747859154,"https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c","Chapter 11  \nSystems of Equations  \nEquation or Function?  \nRemember at the beginning we differentiated between an equation and a function. We said that an equation was essentially sitting there, waiting for the expert mathematician to come along and solve for the unknown value, usually described with an “x”. There was no graph associated with the equation, no height, no function telling the value within the slot how high or how low in the graph to go. We simply solved for x.  \nThe way I like to look at functions is that the height is dependent on what xvalue is in the slot. The function f (x) = x 3 − x has a definite “plan” for what happens to the particular x-value in the slot; you cube it, and then subtract the square root of it, and that’s your height.  \nNow we come to something new that is a little bit of both. I’m going to start with writing what I call a linear equation in both x and y:  \nx − 3y = 5  \nI could easily rewrite this as y = ~~ ~~13~~ ~~ x − ~~ ~~53  \nand make it appear that the y-value  \ndepends on x, like our functions did in previous chapters. But there is something different here in that I could just as easily have written x = 3y + 5 , making x depend on y. either way, we can create a set of coordinate points, put this on a graph, do everything we would do with functions. The only real difference here is that x and y can be said to be equally dependent on the other.  \nBut in the title to this chapter I mention “systems of equations”meaning a group of two or more equations containing the same variables, such as  \n2x − 3y = 6 and x + y = − 1  \nBecause they contain x AND y, I can graph them on a coordinate plane. Perhaps the fastest way to do this is to find the x-intercept and the y-intercept for both:  \nPlug in 0 for x: 2(0) − 3y = 6 says that y = − 2, so (0, − 2) is on the first equation’s line.  \nPlug in 0 for y: 2x − 3(0) = 6 line.  \nPlug in 0 for x: (0) + y = − 1 equation’s line.  \nPlug in 0 for y: x + 0 = − 1 equation’s line.  \nsays that x = 3, so (3,0) is on the first equation’s says that y = − 1, so (0, − 1) is on the second  \nsays that x = − 1 , so (−1,0) is on the second  \nAnd with two points determining a line, we can graph both of these equations:  \nWhat is interesting here to us is that the lines cross. That means that there is a point that both lines share. We call such a point a solution to the system of equations that we have here; that is, this special point “works” for both lines.  \nIt appears that this point, upon close inspection, looks like it could be ( ~~ ~~35~~ ~~ , − 1~~ ~~35~~ ~~ ) or ( ~~ ~~35~~ ~~ , − ~~ ~~85~~ ~~ ). The way to make sure of this, of course, is to plug in these values into each equation to check it out:  \n2( ~~ ~~35~~ ~~ ) − 3( − ~~ ~~85~~ ~~ ) = ~~ ~~65 − − ~~ ~~245 = ~~ ~~305 = 6 , and  \n35 + ( − ~~ ~~85~~ ~~ ) = − ~~ ~~55 = − 1 ,  \nand we see that this coordinate point indeed works for both equations, and is therefore a solution to the system of equations.  \nTypically we write out the system with the equations one above the other, like this:  \n2x + 3y = − 6  \nx + y = − 1  \nThis further shows the “equality” between the two variables here, x and y. One variable is not “more important” or “more dependent” than the other!  \nAn Aside  \nI cannot help mentioning that what we have done above, with the two lines crossing, is a technique to solve any equation.  \nSuppose I needed to know where 3x4 − 5x was equal to 2.7. I would plot y = 3x4 − 5x on a graphing calculator or in DESMOS, and then plot y = 2.7. You’ll remember that the second is a horizontal line, all of whose points have their ycoordinate at 2.7. Where do these two lines cross? That will be the answer I’m looking for.  \nAnd you can see that I have two answers, x = − .502 and x = 1.328; both are approximations of course. Put each of these numbers into our function 3x4 − 5x and you will get (approximately) zero.  \nNow, back to what we were talking about, systems of equations, where the xand y-variables have equ","cbCaicwQk0pjvmEQ","https://ap.wps.com/l/cbCaicwQk0pjvmEQ","pdf",2923113,28,"English","# Equation or Function?\n## Systems of Equations and Graphing\n## Intercepts and Shared Solutions\n## Aside: Solving by Intersections\n## Substitution\n## Elimination","[{\"question\":\"What is the difference between an equation and a function in this chapter?\",\"answer\":\"An equation is treated as something to be solved for the unknown value (often x), without a graph expressing dependence. A function describes how an output value changes based on the input x, such as f(x)=x^3−x determining a height.\"},{\"question\":\"How do you find and graph a system of two equations?\",\"answer\":\"Graph each equation on the coordinate plane by locating intercepts. Substitute x=0 to get the y-intercept and substitute y=0 to get the x-intercept for each line, then use two points to draw each line.\"},{\"question\":\"How does substitution solve a two-variable system?\",\"answer\":\"Solve one equation for one variable in terms of the other, then substitute that expression into the other equation. This reduces the system to a single-variable equation, after which you back-substitute to find the remaining variable.\"}]","Chapter 11 - Systems of Equations - Equation or Function? | PDF",1792132656,71]