ArtsAutosBooksBusinessEducationEntertainmentFamilyFashionFoodGamesGenderHealthHolidaysHomeHubPagesPersonal FinancePetsPoliticsReligionSportsTechnologyTravel

Simultaneous Equations

Updated on July 3, 2016

Simultaneous Equations

As my tutoring students know, I detest calculators. I think the current math curriculum encourages laziness. Working with the numbers themselves gives students more of an idea of how numbers fit together. For example, I often can solve math problems in my head. The reason I can do that is that I did the vast majority of my math problems with pencil and paper or in my head, not with a calculator.

The reason I mention this is that my main tutee (he's in high school and I've been working with him since he was in 2nd grade, so by now he knows how I feel about calculators) wanted to use his calculator, with a simultaneous equation function, to solve the problems. I told him if he wanted my help (which he did) he'd need to learn how to do it without the calculator.

Simultaneous Equations

This picture represents a set of simultaneous equations. These two equations represent two lines on a graph. The solution to the equations is also the point where the two lines intersect.

The Next Step

Multiply the entire top equation by 4 and the entire bottom equation by 3 (so that box equations now have 12x as the first part of thee equation) to get the results in this picture.

After That:

Because out goal is to get this equation down to a solvable one variable equation (and one equation), we will want to subtract the second equation (because the "x" part of both equations will "drop out" when you take 12x from 12x). So we multiply the lower equation by -1 (or, simply, change the plus signs to minus and the minus signs to plus).

Subtract and get this

As you can see from this picture, you subtract each column (x from x, y from y, etc.) and the x drops out leaving only the y and the numbers. Then you solve for y by dividing by 17 and you end up with
y = -2

Once you solve for y, you need to find x:

Substitute -2 for y and this is what you get:

Do the same thing in the other equation:

You should get the same answer for x (this is a way to check your work -- if the answers are the same, then you probably did it right -- if not, you need to redo the problem)

One More Way to do Simultaneous Equations

There is another way to do Simultaneous Equations. That is by using the substitution method -- you solve for X in terms of Y or Y in terms of X.

In this example, we take the second equation (x-y=0) and move the y to the other side by adding it so -- X - Y + Y = 0 + Y. the -Y +Y on the left side cancel out (-Y+Y = 0) and you are left with X=Y.

In the OTHER equation (3x + 2y = -5) and substitute y for x (since y=x, as we just figured out) -- and we get 3y + 2y = -5. Add together and we get 5y=-5 or y = -1. Since x=y, then x=-1. So the answer it (-1, -1).

For the most part, it's harder to do this method, but occasionally, you'll see an equation that makes substitution easier. It's a matter of instinct. While this won't work well most of the time, it's nice to know how to do it when you need it


Compugraph Designs Spoonflower Site

Spoonflower is a place where a designer can design his/her own fabric. I have some designs on fabric there -- click on the picture to see all the fabric designs.

Compugraph Designs Arts Now Site

"Arts Now" is another "Print on Demand" site. They have a nice collection of clocks and watches, including the one pictured here (with math symbols on it). Click on the picture to see the entire site.

Did this help you understand?

    0 of 8192 characters used
    Post Comment

    • srsddn lm profile image

      srsddn lm 

      5 years ago

      It is good not to use calculators while solving problems like this. It sharpens your brain for doing mathematical calculations.

    • Blackspaniel1 profile image


      6 years ago

      My preference is the Gauss Jordan technique.


    This website uses cookies

    As a user in the EEA, your approval is needed on a few things. To provide a better website experience, uses cookies (and other similar technologies) and may collect, process, and share personal data. Please choose which areas of our service you consent to our doing so.

    For more information on managing or withdrawing consents and how we handle data, visit our Privacy Policy at:

    Show Details
    HubPages Device IDThis is used to identify particular browsers or devices when the access the service, and is used for security reasons.
    LoginThis is necessary to sign in to the HubPages Service.
    Google RecaptchaThis is used to prevent bots and spam. (Privacy Policy)
    AkismetThis is used to detect comment spam. (Privacy Policy)
    HubPages Google AnalyticsThis is used to provide data on traffic to our website, all personally identifyable data is anonymized. (Privacy Policy)
    HubPages Traffic PixelThis is used to collect data on traffic to articles and other pages on our site. Unless you are signed in to a HubPages account, all personally identifiable information is anonymized.
    Amazon Web ServicesThis is a cloud services platform that we used to host our service. (Privacy Policy)
    CloudflareThis is a cloud CDN service that we use to efficiently deliver files required for our service to operate such as javascript, cascading style sheets, images, and videos. (Privacy Policy)
    Google Hosted LibrariesJavascript software libraries such as jQuery are loaded at endpoints on the or domains, for performance and efficiency reasons. (Privacy Policy)
    Google Custom SearchThis is feature allows you to search the site. (Privacy Policy)
    Google MapsSome articles have Google Maps embedded in them. (Privacy Policy)
    Google ChartsThis is used to display charts and graphs on articles and the author center. (Privacy Policy)
    Google AdSense Host APIThis service allows you to sign up for or associate a Google AdSense account with HubPages, so that you can earn money from ads on your articles. No data is shared unless you engage with this feature. (Privacy Policy)
    Google YouTubeSome articles have YouTube videos embedded in them. (Privacy Policy)
    VimeoSome articles have Vimeo videos embedded in them. (Privacy Policy)
    PaypalThis is used for a registered author who enrolls in the HubPages Earnings program and requests to be paid via PayPal. No data is shared with Paypal unless you engage with this feature. (Privacy Policy)
    Facebook LoginYou can use this to streamline signing up for, or signing in to your Hubpages account. No data is shared with Facebook unless you engage with this feature. (Privacy Policy)
    MavenThis supports the Maven widget and search functionality. (Privacy Policy)
    Google AdSenseThis is an ad network. (Privacy Policy)
    Google DoubleClickGoogle provides ad serving technology and runs an ad network. (Privacy Policy)
    Index ExchangeThis is an ad network. (Privacy Policy)
    SovrnThis is an ad network. (Privacy Policy)
    Facebook AdsThis is an ad network. (Privacy Policy)
    Amazon Unified Ad MarketplaceThis is an ad network. (Privacy Policy)
    AppNexusThis is an ad network. (Privacy Policy)
    OpenxThis is an ad network. (Privacy Policy)
    Rubicon ProjectThis is an ad network. (Privacy Policy)
    TripleLiftThis is an ad network. (Privacy Policy)
    Say MediaWe partner with Say Media to deliver ad campaigns on our sites. (Privacy Policy)
    Remarketing PixelsWe may use remarketing pixels from advertising networks such as Google AdWords, Bing Ads, and Facebook in order to advertise the HubPages Service to people that have visited our sites.
    Conversion Tracking PixelsWe may use conversion tracking pixels from advertising networks such as Google AdWords, Bing Ads, and Facebook in order to identify when an advertisement has successfully resulted in the desired action, such as signing up for the HubPages Service or publishing an article on the HubPages Service.
    Author Google AnalyticsThis is used to provide traffic data and reports to the authors of articles on the HubPages Service. (Privacy Policy)
    ComscoreComScore is a media measurement and analytics company providing marketing data and analytics to enterprises, media and advertising agencies, and publishers. Non-consent will result in ComScore only processing obfuscated personal data. (Privacy Policy)
    Amazon Tracking PixelSome articles display amazon products as part of the Amazon Affiliate program, this pixel provides traffic statistics for those products (Privacy Policy)