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Sekretari na wageni ofisini

Discussion in 'Jokes/Utani + Udaku/Gossips' started by Mabagala, Dec 12, 2011.

  1. Mabagala

    Mabagala JF-Expert Member

    #1
    Dec 12, 2011
    Joined: Nov 27, 2009
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    Ghafla alichomoza sekretari na kuwatangazia wageni waliopo kwenye benchi wakisubiri kumuona bosi kuwa 'mwenye haja kubwa tu ndo aingie kumuona bosi' kweli kiswahili kipana!
     
  2. Jaluo_Nyeupe

    Jaluo_Nyeupe JF-Expert Member

    #2
    Dec 12, 2011
    Joined: Dec 21, 2010
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    hhahaaaaaaaaahaaaaa! Mi ninahara kabisa sasa sijui haja yangu inaitwaje?
     
  3. Erickb52

    Erickb52 JF-Expert Member

    #3
    Dec 12, 2011
    Joined: Oct 31, 2010
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    hahahaaaaa
     
  4. mgodi

    mgodi JF-Expert Member

    #4
    Dec 12, 2011
    Joined: Nov 5, 2010
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    hapo sijui kama aliinga mtu!
     
  5. tobycow

    tobycow Senior Member

    #5
    Dec 13, 2011
    Joined: Apr 12, 2011
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    hapo secretary angekuwa njemba pangenoga
     
  6. Wa kusoma

    Wa kusoma JF-Expert Member

    #6
    Dec 13, 2011
    Joined: Jul 30, 2011
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    [FONT=&quot]How to solve simultaneous equations by elimination.[/FONT][FONT=&quot][/FONT]​
    [TABLE="class: MsoNormalTable, width: 100"]
    [TR]
    [TD="width: 23%"] [FONT=&quot][​IMG][/FONT][FONT=&quot][/FONT]
    [/TD]
    [TD="width: 77%"] [FONT=&quot]This topic is part of the TCS FREE high school mathematics 'How-to Library'[/FONT][FONT=&quot], and will help you to solve simultaneous equations by elimination.
    [/FONT][FONT=&quot](See the [/FONT][FONT=&quot]index page[/FONT][FONT=&quot] for a list of all available topics in the library.)[/FONT][FONT=&quot] [/FONT][FONT=&quot]To make best use of this topic, you need to download the Algematics software. [/FONT][FONT=&quot]Click here[/FONT][FONT=&quot] for instructions.[/FONT][FONT=&quot][/FONT]
    [/TD]
    [/TR]
    [/TABLE]
    [FONT=&quot][​IMG][/FONT]​
    [FONT=&quot]Theory:[/FONT][FONT=&quot][/FONT]
    [FONT=&quot]In the ‘elimination' method for solving simultaneous equations, two equations are simplified by adding them or subtracting them. This eliminates one of the variables so that the other variable can be found.[/FONT][FONT=&quot][/FONT]
    [FONT=&quot]To add two equations, add the left hand expressions and right hand expressions separately. Similarly, to subtract two equations, subtract the left hand expressions from each other, and subtract the right hand expressions from each other. The following examples will make this clear.[/FONT][FONT=&quot][/FONT]
    [FONT=&quot]Example 1:[/FONT][FONT=&quot] Consider these equations: [/FONT][FONT=&quot][/FONT]
    [FONT=&quot]2x - 5y = 1 [/FONT][FONT=&quot][/FONT]
    [FONT=&quot]3x + 5y = 14 [/FONT][FONT=&quot][/FONT]
    [FONT=&quot]The first equation contains a ‘[/FONT]-[FONT=&quot]5y' term, while the second equation contains a ‘+5y' term. These two terms will cancel if added together, so we will add the equations to eliminate ‘y'.[/FONT][FONT=&quot][/FONT]
    [FONT=&quot]To add the equations, add the left side expressions and the right side expressions separately.[/FONT][FONT=&quot][/FONT]
    [TABLE="class: MsoNormalTable, width: 411"]
    [TR]
    [TD="width: 68"][/TD]
    [TD="width: 293"] [FONT=&quot]2x [/FONT]-[FONT=&quot] 5y =[/FONT][FONT=&quot][/FONT]
    [/TD]
    [TD="width: 83"][/TD]
    [TD="width: 87"] [FONT=&quot]1[/FONT][FONT=&quot][/FONT]
    [/TD]
    [/TR]
    [TR]
    [TD="width: 68"]
    [FONT=&quot]+[/FONT][FONT=&quot] [/FONT][FONT=&quot][/FONT]​
    [/TD]
    [TD="width: 293"] [FONT=&quot]3x + 5y =[/FONT][FONT=&quot][/FONT]
    [/TD]
    [TD="width: 83"]
    [FONT=&quot]+[/FONT][FONT=&quot][/FONT]​
    [/TD]
    [TD="width: 87"] [FONT=&quot]14[/FONT][FONT=&quot][/FONT]
    [/TD]
    [/TR]
    [TR]
    [TD="width: 68"][/TD]
    [TD="width: 293"] [FONT=&quot](2x [/FONT]-[FONT=&quot] 5y) + (3x + 5y) =[/FONT][FONT=&quot][/FONT]
    [/TD]
    [TD="width: 83"][/TD]
    [TD="width: 87"] [FONT=&quot]1 + 14[/FONT][FONT=&quot][/FONT]
    [/TD]
    [/TR]
    [/TABLE]
    [FONT=&quot]Simplifying, [/FONT]-[FONT=&quot]5y and +5y cancel out, so we have:[/FONT][FONT=&quot][/FONT]
    [FONT=&quot]5x = 15 [/FONT][FONT=&quot][/FONT]
    [FONT=&quot]Therefore ‘x' is 3.[/FONT][FONT=&quot][/FONT]
    [FONT=&quot]By substituting 3 for ‘x' into either of the two original equations we can find ‘y'.[/FONT][FONT=&quot][/FONT]
    [FONT=&quot]Example 2: [/FONT][FONT=&quot][/FONT]
    [FONT=&quot]The elimination method will only work if you can eliminate one of the variables by adding or subtracting the equations as in example 1 above. But for many simultaneous equations, this is not the case.[/FONT][FONT=&quot][/FONT]
    [FONT=&quot]For example, consider these equations:[/FONT][FONT=&quot][/FONT]
    [FONT=&quot]2x + 3y = 4 [/FONT][FONT=&quot][/FONT]
    [FONT=&quot] x -[/FONT][FONT=&quot] [/FONT][FONT=&quot]2y = -5 [/FONT][FONT=&quot][/FONT]
    [FONT=&quot]Adding or subtracting these equations will not cancel out the ‘x' or ‘y' terms.[/FONT][FONT=&quot][/FONT]
    [FONT=&quot]Before using the elimination method you may have to multiply every term of one or both of the equations by some number so that equal terms can be eliminated.[/FONT][FONT=&quot][/FONT]
    [FONT=&quot]We could eliminate ‘x' for this example if the second equation had a ‘2x' term instead of an ‘x' term. By multiplying every term in the second equation by 2, the ‘x' term will become ‘2x', like this:[/FONT][FONT=&quot][/FONT]
    [FONT=&quot]
    [/FONT] [FONT=&quot] [/FONT]
    [FONT=&quot]x[/FONT]´[FONT=&quot]2[/FONT][FONT=&quot] [/FONT]-[FONT=&quot] 2y[/FONT]´[FONT=&quot]2[/FONT][FONT=&quot] = [/FONT]-[FONT=&quot]5[/FONT]´[FONT=&quot]2[/FONT][FONT=&quot] [/FONT][FONT=&quot][/FONT]
    [FONT=&quot]giving:[/FONT][FONT=&quot][/FONT]
    [FONT=&quot]2x - 4y = -10 [/FONT][FONT=&quot][/FONT]
    [FONT=&quot]Now the ‘x' term in each equation is the same, and the equations can be subtracted to eliminate ‘x':[/FONT][FONT=&quot][/FONT]
    [TABLE="class: MsoNormalTable, width: 411"]
    [TR]
    [TD="width: 68"][/TD]
    [TD="width: 293"] [FONT=&quot]2x + 3y =[/FONT][FONT=&quot][/FONT]
    [/TD]
    [TD="width: 83"][/TD]
    [TD="width: 87"] [FONT=&quot] 4[/FONT][FONT=&quot][/FONT]
    [/TD]
    [/TR]
    [TR]
    [TD="width: 68"]
    -[FONT=&quot] [/FONT][FONT=&quot][/FONT]​
    [/TD]
    [TD="width: 293"] [FONT=&quot]2x [/FONT]-[FONT=&quot] 4y =[/FONT][FONT=&quot][/FONT]
    [/TD]
    [TD="width: 83"]
    - [FONT=&quot][/FONT]​
    [/TD]
    [TD="width: 87"] -[FONT=&quot]10[/FONT][FONT=&quot][/FONT]
    [/TD]
    [/TR]
    [TR]
    [TD="width: 68"][/TD]
    [TD="width: 293"] [FONT=&quot](2x + 3y) [/FONT]-[FONT=&quot] (2x [/FONT]-[FONT=&quot] 4y) =[/FONT][FONT=&quot][/FONT]
    [/TD]
    [TD="width: 83"][/TD]
    [TD="width: 87"] [FONT=&quot]4 [/FONT]-[FONT=&quot] [/FONT]-[FONT=&quot]10[/FONT][FONT=&quot][/FONT]
    [/TD]
    [/TR]
    [/TABLE]
    [FONT=&quot]Removing the brackets and simplifying, the ‘2x' terms cancel out, so we have:[/FONT][FONT=&quot][/FONT]
    [FONT=&quot]7y = 14 [/FONT][FONT=&quot][/FONT]
    [FONT=&quot]So[/FONT][FONT=&quot][/FONT]
    [FONT=&quot]y[/FONT][FONT=&quot] = 2 [/FONT][FONT=&quot][/FONT]
    [FONT=&quot]The other variable, ‘x', can now be found by substituting 2 for ‘y' into either of the original equations.[/FONT][FONT=&quot][/FONT]
    [TABLE="class: MsoNormalTable, width: 100"]
    [TR]
    [TD="width: 100%"] [FONT=&quot]Sometimes both equations must be modified in order to cancel a variable. For example, to cancel the ‘y' terms for this example, we could multiply the first equation by 4, and the second equation by 3. Then there would be a ‘12y' term in the first equation and a ‘-12y' term in the second equation. Adding the equations would then eliminate ‘y'.[/FONT][FONT=&quot][/FONT]
    [/TD]
    [/TR]
    [/TABLE]
    [FONT=&quot]The steps in the ‘Method' section below demonstrate how to use Algematics to solve simultaneous equations using the elimination method.[/FONT][FONT=&quot][/FONT]
    [TABLE="class: MsoNormalTable, width: 100"]
    [TR]
    [TD="width: 100%"] [FONT=&quot]Download the free support file...[/FONT][FONT=&quot] [/FONT][FONT=&quot]We have created an Algematics document containing the completed example from this topic. It also includes practice exercises to improve your skills.[/FONT][FONT=&quot][/FONT]
    [FONT=&quot]File name:[/FONT][FONT=&quot] 'Simultaneous equations (elimination).alg' File size: 10kb[/FONT][FONT=&quot]
    Click here[/FONT]
    [FONT=&quot] to download the file.[/FONT][FONT=&quot][/FONT]
    [FONT=&quot]If you choose 'Open this file from its current location', then Algematics should open the document immediately. If not, try the other option: 'Save this file to disk', then run Algematics and choose the 'Open' command from the 'File' menu. Locate the saved file and open it. If you do not yet have Algematics installed on your computer, [/FONT][FONT=&quot]click here[/FONT][FONT=&quot] for instructions.[/FONT][FONT=&quot][/FONT]
    [/TD]
    [/TR]
    [/TABLE]
    [FONT=&quot]Method:[/FONT][FONT=&quot][/FONT]
    [FONT=&quot]IMPORTANT[/FONT][FONT=&quot]: This topic assumes that you know how to enter mathematical formulas into Algematics. Find out how by completing the three simple tutorials in the 'Getting Started' section of the Algematics program 'Help'.[/FONT][FONT=&quot][/FONT]
    [FONT=&quot]To solve two equations by the elimination method:[/FONT][FONT=&quot][/FONT]
    [FONT=&quot]-[/FONT][FONT=&quot] [/FONT][FONT=&quot]enter[/FONT][FONT=&quot] the equations,[/FONT][FONT=&quot][/FONT]
    [FONT=&quot]-[/FONT][FONT=&quot] [/FONT][FONT=&quot]prepare[/FONT][FONT=&quot] them if necessary so that one of the variable will cancel,[/FONT][FONT=&quot][/FONT]
    [FONT=&quot]-[/FONT][FONT=&quot] [/FONT][FONT=&quot]add or subtract[/FONT][FONT=&quot] them to eliminate one of the variables, then[/FONT][FONT=&quot][/FONT]
    [FONT=&quot]- substitute to find the second variable.[/FONT][FONT=&quot][/FONT]
    [FONT=&quot]We will use example 2 from the ‘Theory' section above in this demonstration.[/FONT][FONT=&quot][/FONT]​
    [FONT=&quot] Step 1 Enter the equations to solve[/FONT][FONT=&quot][/FONT]
    [FONT=&quot]Click [​IMG] and type your first equation into the maths box in the data entry dialog box. [/FONT][FONT=&quot][/FONT]
    [FONT=&quot]If the ‘EMPTY' message is not displayed between the blue buttons, click the [​IMG] button until the message: ‘INSERT' appears. [/FONT][FONT=&quot][/FONT]
    [FONT=&quot] Maths... [/FONT][FONT=&quot][/FONT]
    [FONT=&quot] [/FONT][FONT=&quot] 2x + 3y = 4 [/FONT][FONT=&quot][/FONT]
    [FONT=&quot] [/FONT][FONT=&quot][/FONT]
    [FONT=&quot] [/FONT][FONT=&quot][/FONT]
    [FONT=&quot]Click the [​IMG] button, then type the second equation into the maths box:[/FONT][FONT=&quot][/FONT]
    [FONT=&quot] Maths... [/FONT][FONT=&quot][/FONT]
    [FONT=&quot] [/FONT][FONT=&quot] x – 2y = -5 [/FONT][FONT=&quot][/FONT]
    [FONT=&quot] [/FONT][FONT=&quot][/FONT]
    [FONT=&quot] [/FONT][FONT=&quot][/FONT]
    [FONT=&quot] Check the ‘Rule Of Line ...' check box so that the two initial equations will stand out at the top of all the working.[/FONT][FONT=&quot][/FONT]
    [FONT=&quot]Click [​IMG][/FONT][FONT=&quot][/FONT]
    [FONT=&quot]Step 2 [/FONT][FONT=&quot]If necessary[/FONT][FONT=&quot], prepare either or both equations for eliminating a variable.[/FONT][FONT=&quot][/FONT]
    [FONT=&quot]Decide which variable you are going to eliminate from the equations and whether you are going to add the equations or subtract them.[/FONT][FONT=&quot][/FONT]
    [FONT=&quot]For the example equations, [/FONT][FONT=&quot][/FONT]
    [FONT=&quot]2x + 3y = 4 [/FONT][FONT=&quot][/FONT]
    [FONT=&quot] x - 2y = -5 [/FONT][FONT=&quot][/FONT]
    [FONT=&quot]we will multiply the second equation through by 2 and eliminate variable ‘x' by subtracting the second equation from the first. [/FONT][FONT=&quot][/FONT]
    [FONT=&quot]To modify an equation, use a copy of the original equation.[/FONT][FONT=&quot][/FONT]
    [FONT=&quot]To copy an equation[/FONT][FONT=&quot], hold down the Ctrl key on the keyboard while you click on the equation with the mouse and drag it down below the others. The copy will appear when you release the mouse button.[/FONT][FONT=&quot] [/FONT][FONT=&quot][/FONT]
    [FONT=&quot]For the example, copy the equation: [/FONT][FONT=&quot][/FONT]
    [FONT=&quot]x[/FONT][FONT=&quot] [/FONT]-[FONT=&quot] 2y = [/FONT]-[FONT=&quot]5 [/FONT][FONT=&quot][/FONT]
    [FONT=&quot]below the others, so that the display looks like this:
    (The newly copied equation is shown here in red)[/FONT][FONT=&quot][/FONT]
    [FONT=&quot]2x + 3y = 4 [/FONT][FONT=&quot][/FONT]
    [FONT=&quot] x[/FONT][FONT=&quot] [/FONT]-[FONT=&quot] 2y = [/FONT]-[FONT=&quot]5[/FONT][FONT=&quot][/FONT]

    [FONT=&quot] x[/FONT][FONT=&quot] [/FONT]-[FONT=&quot] 2y = [/FONT]-[FONT=&quot]5[/FONT][FONT=&quot] [/FONT][FONT=&quot][/FONT]
    [FONT=&quot]To multiply the bottom equation by 2:[/FONT][FONT=&quot][/FONT]
    [FONT=&quot]Click on the input box and type ‘2': [/FONT][FONT=&quot][/FONT]
    [TABLE="class: MsoNormalTable, width: 276"]
    [TR]
    [TD="width: 151"]
    [FONT=&quot]Input[/FONT][FONT=&quot] [/FONT][FONT=&quot][/FONT]​
    [/TD]
    [TD="width: 207"] [FONT=&quot] 2[/FONT][FONT=&quot][/FONT]

    [/TD]
    [/TR]
    [/TABLE]
    [FONT=&quot]Click [​IMG][/FONT][FONT=&quot][/FONT]
    [FONT=&quot]Multiplying by '2', both sides ...[/FONT][FONT=&quot] [/FONT][FONT=&quot][/FONT]
    [FONT=&quot](x - 2y2 = -5´2 [/FONT][FONT=&quot][/FONT]
    [FONT=&quot]Click [​IMG] (expand, this removes the brackets)[/FONT][FONT=&quot][/FONT]
    [FONT=&quot]Expanding, both sides ...[/FONT][FONT=&quot] [/FONT][FONT=&quot][/FONT]
    [FONT=&quot]x[/FONT]´[FONT=&quot]2 [/FONT]-[FONT=&quot] 2y[/FONT]´[FONT=&quot]2 = [/FONT]-[FONT=&quot]5[/FONT]´[FONT=&quot]2 [/FONT][FONT=&quot][/FONT]
    [FONT=&quot]Click [​IMG] (simplify all)[/FONT][FONT=&quot][/FONT]
    [FONT=&quot]Simplifying, both sides ...[/FONT][FONT=&quot] [/FONT][FONT=&quot][/FONT]
    [FONT=&quot]2x - 4y = -10 [/FONT][FONT=&quot][/FONT]
    [FONT=&quot]RULE OFF[/FONT][FONT=&quot]: Double click the mouse on this last data set to open the data entry dialog box. Check the ‘Rule Off Line ...' check box so that the two initial equations will stand out at the top of all the working.[/FONT][FONT=&quot][/FONT]

    [FONT=&quot]Click [​IMG][/FONT][FONT=&quot][/FONT]
    [FONT=&quot]NOTE[/FONT][FONT=&quot]: If you need to modify the other equation as well, copy it to the bottom and follow the same procedure, ending with another rule off line.[/FONT][FONT=&quot][/FONT]

    [FONT=&quot]Step 3 Eliminate the first variable[/FONT][FONT=&quot][/FONT]
    [FONT=&quot]Copy the two equations that you are going to add or subtract to the bottom of the working page, using the copying technique described above.[/FONT][FONT=&quot][/FONT]
    [FONT=&quot]For the example, the last few lines of the display will now look like this: [/FONT][FONT=&quot][/FONT]
    [FONT=&quot]Simplifying, both sides ... [/FONT][FONT=&quot][/FONT]
    [FONT=&quot]2x - 4y = -10 [/FONT][FONT=&quot][/FONT]

    [FONT=&quot]2x - 4y = -10 [/FONT][FONT=&quot][/FONT]
    [FONT=&quot]2x + 3y = 4 [/FONT][FONT=&quot][/FONT]
    [FONT=&quot] [/FONT][FONT=&quot] [/FONT][FONT=&quot]Click on the bottom equation to make it the target.[/FONT][FONT=&quot][/FONT]
    [TABLE="class: MsoNormalTable, width: 213"]
    [TR]
    [TD="width: 65"][/TD]
    [TD="width: 209"] [FONT=&quot] 2x + 3y = 4[/FONT][FONT=&quot][/FONT]
    [/TD]
    [/TR]
    [/TABLE]
    [FONT=&quot] [/FONT][FONT=&quot][/FONT]
    [FONT=&quot]Click on the input box arrow and select the other equation. [/FONT][FONT=&quot][/FONT]
    [FONT=&quot]Click here to select[/FONT][FONT=&quot][/FONT]
    [FONT=&quot]For the example, select: ‘2x – 4y = -10' in the input box.[/FONT][FONT=&quot][/FONT]
    [TABLE="class: MsoNormalTable, width: 276"]
    [TR]
    [TD="width: 151"]
    [FONT=&quot]Input[/FONT][FONT=&quot] [/FONT][FONT=&quot][/FONT]​
    [/TD]
    [TD="width: 207"] [FONT=&quot] [/FONT][FONT=&quot]2x – 4y = -10[/FONT][FONT=&quot][/FONT]

    [/TD]
    [/TR]
    [/TABLE]
    [FONT=&quot]Click [​IMG] to add the equations, or [​IMG] to subtract them.
    (For the example, we are subtracting the equations to eliminate ‘2x', so click [​IMG] )[/FONT][FONT=&quot][/FONT]
    [FONT=&quot]Click [​IMG] (simplify all). You will now have an equation in only one variable.
    [/FONT][FONT=&quot][/FONT]
    [TABLE="class: MsoNormalTable, width: 100"]
    [TR]
    [TD="width: 100%"] [FONT=&quot]For the example, the last few lines of the display will now look like this: [/FONT][FONT=&quot][/FONT]
    [FONT=&quot]2x - 4y = -10 [/FONT][FONT=&quot][/FONT]
    [FONT=&quot]2x + 3y = 4 [/FONT][FONT=&quot][/FONT]
    [FONT=&quot]Subtracting '2x - 4y = -10' ... [/FONT][FONT=&quot][/FONT]
    [FONT=&quot](2x + 3y) - (2x - 4y) = 4 -[/FONT][FONT=&quot] [/FONT][FONT=&quot]-10 [/FONT][FONT=&quot][/FONT]
    [FONT=&quot]Simplifying, both sides ... [/FONT][FONT=&quot][/FONT]
    [FONT=&quot]7y = 14 [/FONT]ï[FONT=&quot] [/FONT][FONT=&quot]Solve this equation to find 'y'.[/FONT][FONT=&quot][/FONT]
    [/TD]
    [/TR]
    [/TABLE]
    [FONT=&quot]For the example, we just divide both sides by 7 and simplify. Click on the input box, type 7, then click [​IMG], then [​IMG]. This gives the result: [/FONT][FONT=&quot][/FONT]
    [FONT=&quot]y[/FONT][FONT=&quot] = 2 [/FONT][FONT=&quot][/FONT]
    [FONT=&quot]Make a rule off line under this last step as explained above.[/FONT]
    [FONT=&quot]Step 4 Substitute to find the other variable[/FONT][FONT=&quot][/FONT]
    [FONT=&quot]Copy either of the original equations to the bottom of the working page, using the copying technique described earlier in this article.[/FONT]
    [FONT=&quot]For the example, we will copy the equation: [/FONT][FONT=&quot][/FONT]
    [FONT=&quot]x[/FONT][FONT=&quot] [/FONT]-[FONT=&quot] 2y = [/FONT]-[FONT=&quot]5 [/FONT][FONT=&quot][/FONT]
    [FONT=&quot]Click on the input box, and type or select the equation for the known variable.[/FONT]
    [FONT=&quot]For the example, select or type: y = 2 [/FONT][FONT=&quot][/FONT]
    [TABLE="class: MsoNormalTable, width: 276"]
    [TR]
    [TD="width: 151"]
    [FONT=&quot]Input[/FONT][FONT=&quot] [/FONT][FONT=&quot][/FONT]​
    [/TD]
    [TD="width: 207"] [FONT=&quot] [/FONT][FONT=&quot]y = 2[/FONT][FONT=&quot][/FONT]

    [/TD]
    [/TR]
    [/TABLE]
    [FONT=&quot]Click [​IMG] (substitute) and then [​IMG] (simplify all).[/FONT]
    [FONT=&quot]You will now have an equation in the other variable. Solve this equation to find the value of this other variable.[/FONT]
    [FONT=&quot]For the example, the equation is: [/FONT][FONT=&quot][/FONT]
    [FONT=&quot]x [/FONT]-[FONT=&quot] 4 = [/FONT]-[FONT=&quot]5[/FONT][FONT=&quot][/FONT]
    [FONT=&quot]In this case, we add 4 to both sides and simplify:
    Click on the input box, type 4,click [​IMG], then [​IMG]. This gives the result: [/FONT]
    [FONT=&quot][/FONT]
    [FONT=&quot]x [/FONT][FONT=&quot] = [/FONT]-[FONT=&quot]1 [/FONT][FONT=&quot][/FONT]
    [FONT=&quot]Step 5 [OPTIONAL] Validate the solutions by substitution[/FONT][FONT=&quot][/FONT]
    [FONT=&quot]You can test your answers to make sure they are correct. Click on the input box, and type both solutions with a comma between them:[/FONT]
    [FONT=&quot]For the example, type: x = -1, y = 2 [/FONT][FONT=&quot][/FONT]
    [TABLE="class: MsoNormalTable, width: 276"]
    [TR]
    [TD="width: 151"]
    [FONT=&quot]Input[/FONT][FONT=&quot] [/FONT][FONT=&quot][/FONT]​
    [/TD]
    [TD="width: 207"] [FONT=&quot] [/FONT][FONT=&quot]x = -1, y = 2[/FONT][FONT=&quot][/FONT]

    [/TD]
    [/TR]
    [/TABLE]
    [FONT=&quot]Click on one of the original equations to make it the target data set[/FONT][FONT=&quot][/FONT]
    [FONT=&quot]2x + 3y = 4 [/FONT][FONT=&quot][/FONT]
    [FONT=&quot]Click [​IMG] (substitute)[/FONT][FONT=&quot][/FONT]
    [FONT=&quot]2´-1 + 3´2 = 4 [/FONT][FONT=&quot][/FONT]
    [FONT=&quot]Click [​IMG] (simplify all)[/FONT][FONT=&quot][/FONT]
    [FONT=&quot]4 = 4 [/FONT][FONT=&quot][/FONT]
    [FONT=&quot]Click on the other original equation to make it the target data set[/FONT][FONT=&quot][/FONT]
    [FONT=&quot]x[/FONT][FONT=&quot] [/FONT]-[FONT=&quot] 2y = [/FONT]-[FONT=&quot]5 [/FONT][FONT=&quot][/FONT]
    [FONT=&quot]Click [​IMG] (substitute)[/FONT][FONT=&quot][/FONT]
    [FONT=&quot]-1 - 2´2 = -5 [/FONT][FONT=&quot][/FONT]
    [FONT=&quot]Click [​IMG] (simplify all)[/FONT][FONT=&quot][/FONT]
    [FONT=&quot]-5 = -5 [/FONT][FONT=&quot][/FONT]
    [FONT=&quot]If you get equations that are true in both cases, then your solutions are correct ! [/FONT][FONT=&quot][/FONT]
    [FONT=&quot]For the example, we get these equations: [/FONT][FONT=&quot][/FONT]
    [FONT=&quot]-5 = -5 [/FONT][FONT=&quot][/FONT]
    [FONT=&quot]and[/FONT][FONT=&quot][/FONT]
    [FONT=&quot] 4 = 4 [/FONT][FONT=&quot][/FONT]
    [FONT=&quot]so the solutions: x = [/FONT]-[FONT=&quot]1, y = 2 are correct. [/FONT][FONT=&quot][/FONT]
    [FONT=&quot] [/FONT][FONT=&quot][/FONT]
    [TABLE="class: MsoNormalTable, width: 100"]
    [TR]
    [TD="width: 100%"]
    [FONT=&quot]Still don't understand or have further questions about this topic ?
    [/FONT]
    [FONT=&quot]Then ask us! [/FONT][FONT=&quot]Click here[/FONT][FONT=&quot] now![/FONT][FONT=&quot][/FONT]​
    [/TD]
    [/TR]
    [/TABLE]
    [FONT=&quot] [/FONT]
     
  7. Viol

    Viol JF-Expert Member

    #7
    Dec 13, 2011
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    naona unajitahidi kutafuta bwana
     
  8. Wa kusoma

    Wa kusoma JF-Expert Member

    #8
    Dec 13, 2011
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    Pata darasa la bure
     
  9. L

    Luluka JF-Expert Member

    #9
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    kuna mtu anawashwa masaburi huku ndani!msaidieni jamani.anaitwa w.a.k.u.s.o.m.a
     
  10. Mabagala

    Mabagala JF-Expert Member

    #10
    Dec 13, 2011
    Joined: Nov 27, 2009
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    huyu wa kusoma atakuwa na yeye aende kwa bosi naona ana haja kubwa ya kusaidiwa
     
  11. Viol

    Viol JF-Expert Member

    #11
    Dec 13, 2011
    Joined: Dec 15, 2009
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    mambo yake si mazuri
     
  12. Viol

    Viol JF-Expert Member

    #12
    Dec 13, 2011
    Joined: Dec 15, 2009
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    huyo hasaidiki
     
  13. stevoh

    stevoh JF-Expert Member

    #13
    Dec 14, 2011
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    naona hajaangalia ameingia kwenye forum gan kwan kachanganya utan na masomo wap na wap?
     
  14. Angel Nylon

    Angel Nylon JF-Expert Member

    #14
    Dec 14, 2011
    Joined: Jul 26, 2011
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    Hee, makubwa! Naona kama ww ndo uliekua na haja kubwa ya kumuona boss. Tn mpk makalio yyalikua yanawasha......
     
  15. msani

    msani JF-Expert Member

    #15
    Dec 14, 2011
    Joined: Jun 13, 2011
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    mlio quote hapo nanyi mas****uri tuu
     
  16. Katavi

    Katavi Platinum Member

    #16
    Dec 14, 2011
    Joined: Aug 31, 2009
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    Ha ha haah!! Humu ndani full bhangi.
     
  17. Wa kusoma

    Wa kusoma JF-Expert Member

    #17
    Dec 14, 2011
    Joined: Jul 30, 2011
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    Tatizo lenu ni nini?
     
  18. Jakubumba

    Jakubumba JF-Expert Member

    #18
    Dec 14, 2011
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    bange zako, yani nitakula na wewe Sahani moja daima
     
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