{"branchingScenario":{"content":[{"type":{"library":"H5P.AdvancedText 1.1","params":{"text":"

Interpreting 3 \u00d7 3 systems of equations<\/h2>\n\n
A 3 \u00d7 3 system of equations is one that is made up of three equations with three different variables. Each equation forms a plane in 3D space. The ways in which these planes interact, or don't, will determine the number of possible solutions for the system. There are clues in the equations that will help you visualise the geometric interpretations and the number of solutions.<\/p>\n\n
Let's have a go.<\/p>\n"},"subContentId":"dfb980a5-851f-4bd2-9d0e-bf7f653b1715","metadata":{"contentType":"Text","license":"U","title":"node-9850_3x3 Braching scenario_Question"}},"showContentTitle":false,"forceContentFinished":"useBehavioural","feedback":{"subtitle":""},"contentBehaviour":"useBehavioural","nextContentId":1},{"type":{"library":"H5P.BranchingQuestion 1.0","params":{"branchingQuestion":{"alternatives":[{"nextContentId":3,"feedback":{"title":"
Correct!<\/p>\n","subtitle":"
There is no pattern in the coefficients or constants, so these planes must intersect at a single point.<\/p>\n","endScreenScore":2},"text":"Three planes that intersect at a single point"},{"nextContentId":2,"feedback":{"title":"
Not quite.<\/p>\n","subtitle":"
It looks like you need to refresh your memory about how to determine the geometric interpretation of a system of equations.<\/p>\n","endScreenScore":-1},"text":"Three parallel planes"},{"nextContentId":2,"feedback":{"title":"
Whoops, that's not right.<\/p>\n","subtitle":"
It looks like you need to refresh your memory about how to determine the geometric interpretation of a system of equations.<\/p>\n","endScreenScore":-1},"text":"Two parallel planes that are intersected by a third plane"},{"nextContentId":2,"feedback":{"title":"
Not quite.<\/p>\n","subtitle":"
It looks like you need to refresh your memory about how to determine the geometric interpretation of a system of equations.<\/p>\n","endScreenScore":-1},"text":"Three planes that all intersect along a common line"}],"question":"

Q1a Consider the following system of equations:<\/h3>\n\n
2x<\/em> - y<\/em> + z<\/em> = 5<\/p>\n\n
x<\/em> + y<\/em> + 2z<\/em> = 7<\/p>\n\n
3x<\/em> + 4y<\/em> - z<\/em> = 10<\/p>\n\n
Which one of the following best describes the geometric interpretation of the relationship between the three planes?<\/p>\n"}},"subContentId":"b86265b2-5130-4202-98fd-974b5d5314ef","metadata":{"contentType":"Branching Question","license":"U","title":"Geometric interpretations"}},"showContentTitle":false,"forceContentFinished":"useBehavioural","feedback":{"subtitle":""},"contentBehaviour":"enabled"},{"type":{"library":"H5P.AdvancedText 1.1","params":{"text":"

Geometric interpretations<\/h3>\n\n
When determining the relationship between three planes, consider if there are any patterns between the coefficients and constants in the three equations. The more closely related the coefficients and constants in the equations, the more similar position of the planes. For example, if the same pattern applies to all coefficients and constants, then the planes are coincident, while zero patterns between coefficients and constants mean the planes intersect at a single point.<\/p>\n\n
Now, let's try a different question!<\/p>\n"},"subContentId":"c4f8816c-2d5c-4d5d-aa0e-a19df1bf59d5","metadata":{"contentType":"Text","license":"U","title":"Geometric interpretations clues 1"}},"showContentTitle":false,"forceContentFinished":"useBehavioural","feedback":{"title":"
Have another go!<\/p>\n","subtitle":""},"contentBehaviour":"useBehavioural","nextContentId":3},{"type":{"library":"H5P.BranchingQuestion 1.0","params":{"branchingQuestion":{"alternatives":[{"nextContentId":5,"feedback":{"title":"
Incorrect.<\/p>\n","subtitle":"
It looks like you need to refresh your memory about how to determine the geometric interpretation of a system of equations.<\/p>\n","endScreenScore":-1},"text":"Three parallel planes"},{"nextContentId":5,"feedback":{"title":"
Incorrect.<\/p>\n","subtitle":"
It looks like you need to refresh your memory about how to determine the geometric interpretation of a system of equations.<\/p>\n","endScreenScore":-1},"text":"Three planes where the intersection between each pair of planes is parallel to the third plane"},{"nextContentId":4,"feedback":{"title":"
Good job!<\/p>\n","subtitle":"
There is one linear relationship between all coefficients and constants, so these planes must completely overlap.<\/p>\n","endScreenScore":2},"text":"Three coincident planes"},{"nextContentId":5,"feedback":{"title":"
Incorrect.<\/p>\n","subtitle":"
It looks like you need to refresh your memory about how to determine the geometric interpretation of a system of equations.<\/p>\n","endScreenScore":-1},"text":"Three planes that intersect at a single point"}],"question":"
Q1b Consider the following system of equations:<\/h3>\n\n
6x<\/em> + 3y<\/em> - 9z<\/em> = 18<\/p>\n\n
2x<\/em> + y<\/em> - 3z<\/em> = 6<\/p>\n\n
-4x<\/em> - 2y<\/em> + 6z<\/em> = -12<\/p>\n\n
Which one of the following best describes the geometric interpretation of the relationship between the three planes?<\/p>\n"}},"subContentId":"402f4a4b-d398-4091-8722-ad0be13f0be5","metadata":{"contentType":"Branching Question","license":"U","title":"Q1b"}},"showContentTitle":false,"forceContentFinished":"useBehavioural","feedback":{"subtitle":""},"contentBehaviour":"useBehavioural"},{"type":{"library":"H5P.BranchingQuestion 1.0","params":{"branchingQuestion":{"alternatives":[{"nextContentId":-1,"feedback":{"title":"
Incorrect.<\/p>\n","subtitle":"
It looks like you find geometric interpretations challenging. Contact your teacher and let them know you need some help.<\/p>\n","endScreenScore":-1},"text":"Three planes that intersect at a single point"},{"nextContentId":-1,"feedback":{"title":"
Incorrect.<\/p>\n","subtitle":"
It looks like you find geometric interpretations challenging. Contact your teacher and let them know you need some help.<\/p>\n","endScreenScore":-1},"text":"Three coincident planes"},{"nextContentId":-1,"feedback":{"subtitle":"
It looks like you find geometric interpretations challenging. Contact your teacher and let them know you need some help.<\/p>\n","title":"
Whoops! That's not right.<\/p>\n","endScreenScore":-1},"text":"Three planes where the intersection between each pair of planes is parallel to the third plane"},{"nextContentId":6,"feedback":{"title":"
Well done!<\/p>\n","subtitle":"
There is a linear relationship between the coefficients of all three equations, but the constants do not follow the same pattern. These planes must be parallel.<\/p>\n","endScreenScore":2},"text":"Three parallel planes"}],"question":"
Q1c Consider the following system of equations:<\/h3>\n\n
2x<\/em> - 2y<\/em> + 3z<\/em> = 5<\/p>\n\n
-2x<\/em> + 2y<\/em> - 3z<\/em> = 7<\/p>\n\n
4x<\/em> - 4y<\/em> + 6z<\/em> = 17<\/p>\n\n
Which one of the following best describes the geometric interpretation of the relationship between the three planes?<\/p>\n"}},"subContentId":"059b3185-f32e-4048-9b89-ade128bc7c50","metadata":{"contentType":"Branching Question","license":"U","title":"Q1c"}},"showContentTitle":false,"forceContentFinished":"useBehavioural","feedback":{"subtitle":""},"contentBehaviour":"useBehavioural"},{"type":{"library":"H5P.AdvancedText 1.1","params":{"text":"
Geometric interpretations<\/h3>\n\n
When determining the relationship between three planes, consider the coefficients and constants in the three equations.<\/p>\n\n
\n\t
If there is a linear pattern between all coefficients and constants, then the three planes are coincident and completely overlap.<\/li>\n\t
If you can add two of the equations together to produce the third, then the three planes all intersect along a common line.<\/li>\n\t
If there is a linear relationship between the coefficients, but the constants do not fit the same pattern, then the three planes are parallel.<\/li>\n\t
If there is a linear relationship between the coefficients of two equations but not the third equation, and the constants do not fit the same pattern, then two of the planes are parallel and intersected by the third.<\/li>\n\t
If, when comparing any two equations, there is a linear relationship between all three coefficients, but the linear relationship is different between each pair of equations, then each plane is parallel to the intersection of the other two planes.<\/li>\n\t
If there is no pattern between any of the coefficients or constants between the equations, then the planes intersect at a single point. <\/li>\n<\/ol>\n\n
Now, let's try a different question!<\/p>\n"},"subContentId":"07948630-7eb2-4220-957b-15de95281cf8","metadata":{"contentType":"Text","license":"U","title":"Geometric interpretations clues 2"}},"showContentTitle":false,"forceContentFinished":"useBehavioural","feedback":{"subtitle":""},"contentBehaviour":"useBehavioural","nextContentId":4},{"type":{"library":"H5P.BranchingQuestion 1.0","params":{"branchingQuestion":{"alternatives":[{"nextContentId":9,"feedback":{"title":"
Incorrect.<\/p>\n","subtitle":"
Revise what makes equations consistent<\/strong> or inconsistent <\/strong>and dependent <\/strong>or independent<\/strong>.<\/p>\n","endScreenScore":-1},"text":"Consistent and dependent"},{"nextContentId":7,"feedback":{"title":"
That's right!<\/p>\n","subtitle":"
no matter how you solve this system, you will consistently <\/strong>reach the same solution and each equation provides new informtion about the system and so are independent <\/strong>of each other.<\/p>\n","endScreenScore":2},"text":"Consistent and independent"},{"nextContentId":9,"feedback":{"title":"
Incorrect.<\/p>\n","subtitle":"
Revise what makes equations consistent<\/strong> or inconsistent <\/strong>and dependent <\/strong>or independent<\/strong>.<\/p>\n","endScreenScore":-1},"text":"Inconsistent and dependent"},{"nextContentId":9,"feedback":{"title":"
Incorrect.<\/p>\n","subtitle":"
Revise what makes equations consistent<\/strong> or inconsistent <\/strong>and dependent <\/strong>or independent<\/strong>.<\/p>\n","endScreenScore":-1},"text":"Inconsistent and independent"}],"question":"
2a Consider the following system of equations.<\/h3>\n\n
2x<\/em> - y<\/em> + z<\/em> = 5<\/p>\n\n
x<\/em> + y<\/em> + 2z<\/em> = 7<\/p>\n\n
3x<\/em> + 4y<\/em> - z<\/em> = 10<\/p>\n\n
Which of hte following pairs of terms best describes these equations?<\/p>\n"}},"subContentId":"e1ff99af-7270-44c0-b1b1-e1148e8aa284","metadata":{"contentType":"Branching Question","license":"U","title":"Q2a"}},"showContentTitle":false,"forceContentFinished":"useBehavioural","feedback":{"subtitle":""},"contentBehaviour":"useBehavioural"},{"type":{"library":"H5P.BranchingQuestion 1.0","params":{"branchingQuestion":{"alternatives":[{"nextContentId":8,"feedback":{"title":"
That's the one!<\/p>\n","subtitle":"
Wen you solve this system of equation, you will consistently<\/strong> achieve the same solution. However, because the coefficenty forma liner patten, they are dependent <\/strong>on each other.<\/p>\n","endScreenScore":2},"text":"Consistent and dependent"},{"nextContentId":10,"feedback":{"title":"
Incorrect.<\/p>\n","subtitle":"
Revise the terms used to describe how the equations will behave when solving the system.<\/p>\n","endScreenScore":-1},"text":"Consistent and independent"},{"nextContentId":10,"feedback":{"title":"
Incorrect.<\/p>\n","subtitle":"
Revise the terms used to describe how the equations will behave when solving the system.<\/p>\n","endScreenScore":-1},"text":"Inconsistent and a contradiction"},{"nextContentId":10,"feedback":{"title":"
Incorrect.<\/p>\n","subtitle":"
Revise the terms used to describe how the equations will behave when solving the system.<\/p>\n","endScreenScore":-1},"text":"Inconsistent and independent"}],"question":"
2b Consider the following system of equations.<\/h3>\n\n
6x<\/em> + 3y<\/em> - 9z<\/em> = 18<\/p>\n\n
2x<\/em> + y<\/em> - 3z<\/em> = 6<\/p>\n\n
-4x<\/em> - 2y<\/em> + 6z<\/em> = -12<\/p>\n\n
Which of the following pairs of terms best describes these equations?<\/p>\n"}},"metadata":{"contentType":"Branching Question","license":"U","title":"Q2b"},"subContentId":"89bf8684-b3ba-4c5a-af09-5599432d52e6"},"showContentTitle":false,"forceContentFinished":"useBehavioural","feedback":{"subtitle":""},"contentBehaviour":"useBehavioural"},{"type":{"library":"H5P.BranchingQuestion 1.0","params":{"branchingQuestion":{"alternatives":[{"nextContentId":-1,"feedback":{"title":"
Incorrect.<\/p>\n","subtitle":"
It looks like you find describing systems of equations challenging. Contact your teacher and let them know you need some help.<\/p>\n","endScreenScore":-1},"text":"Consistent and dependent"},{"nextContentId":-1,"feedback":{"title":"
Incorrect.<\/p>\n","subtitle":"
It looks like you find describing systems of equations challenging. Contact your teacher and let them know you need some help.<\/p>\n","endScreenScore":-1},"text":"Consistent and independent"},{"nextContentId":11,"feedback":{"title":"
Well done!<\/p>\n","subtitle":"
Correct! There is no solution for this system that will satisfy all the variables at the same time - inconsistent<\/strong>, and will reduce to 0 = n, which is a contradiction<\/strong>.<\/p>\n","endScreenScore":2},"text":"Inconsistent and a contradiction"},{"nextContentId":-1,"feedback":{"title":"
Incorrect.<\/p>\n","subtitle":"
It looks like you find describing systems of equations challenging. Contact your teacher and let them know you need some help.<\/p>\n","endScreenScore":-1},"text":"Inconsistent and independent"}],"question":"
2c Consider the following system of equations.<\/h3>\n\n
2x<\/em> - 2y<\/em> + 3z<\/em> = 5<\/p>\n\n
-2x<\/em> + 2y<\/em> - 3z<\/em> = 7<\/p>\n\n
4x<\/em> - 4y<\/em> + 6z<\/em> = 17<\/p>\n\n
Which of the following pairs of terms best describes these equations?<\/p>\n"}},"metadata":{"contentType":"Branching Question","license":"U","title":"Q2c"},"subContentId":"4f16efb8-8fe3-4d82-a199-2f7c582c95fd"},"showContentTitle":false,"forceContentFinished":"useBehavioural","feedback":{"subtitle":""},"contentBehaviour":"useBehavioural"},{"type":{"library":"H5P.AdvancedText 1.1","params":{"text":"
Ways of describing systems of equations<\/h3>\n\n
The ways in which the equations are related in a system will affect how you can or can't solve them. The following terms are used to describe systems of equations. <\/p>\n\n
\n\t
Independent <\/strong>equations show no patterns between coefficients or constants. In other words, no equation can be derived from another. <\/li>\n\t
Dependent <\/strong>equations show linear patterns between the coefficients and constants, which means the equations can be reduced to 0 = 0.<\/li>\n\t
Consistent <\/strong>equations will always result in the same solution, no matter the techniques you use to solve them (there is often more than one way to solve simultaneous equations).<\/li>\n\t
Inconsistent <\/strong>equations do not have a common solution that makes all equations true simultaneously because they will reduce to 0 = n. This means they are contradictory<\/strong> - not mathematically true.<\/li>\n<\/ul>\n\n
OK, then. Let's have another go!<\/p>\n"},"subContentId":"7ad7a37d-1b99-4f05-ab50-5d245b3135a2","metadata":{"contentType":"Text","license":"U","title":"Terms describing systems"}},"showContentTitle":false,"forceContentFinished":"useBehavioural","feedback":{"subtitle":""},"contentBehaviour":"useBehavioural","nextContentId":7},{"type":{"library":"H5P.AdvancedText 1.1","params":{"text":"
Ways of describing systems of equations<\/h3>\n\n
The ways in which the equations are related in a system will affect how you can or can't solve them. The following terms are used to describe systems of equations. <\/p>\n\n
\n\t
Independent <\/strong>equations show no patterns between coefficients or constants. In other words, no equation can be derived from another. <\/li>\n\t
Dependent <\/strong>equations show linear patterns between the coefficients and constants, which means the equations can be reduced to 0 = 0.<\/li>\n\t
Consistent <\/strong>equations will always result in the same solution, no matter the techniques you use to solve them (there is often more than one way to solve simultaneous equations).<\/li>\n\t
Inconsistent <\/strong>equations do not have a common solution that makes all equations true simultaneously because they will reduce to 0 = n. This means they are contradictory <\/strong>- not mathematically true.<\/li>\n<\/ul>\n\n
OK, then. Let's try one more time.<\/p>\n"},"metadata":{"contentType":"Text","license":"U","title":"Terms describing systems"},"subContentId":"c2a20274-c58b-4e68-a07a-7ae76a00c423"},"showContentTitle":false,"forceContentFinished":"useBehavioural","feedback":{"subtitle":""},"contentBehaviour":"useBehavioural","nextContentId":8},{"type":{"library":"H5P.BranchingQuestion 1.0","params":{"branchingQuestion":{"alternatives":[{"nextContentId":12,"feedback":{"title":"
Correct!<\/p>\n","subtitle":"
Consistent and indepedent systems of equations will have a unique solution.<\/p>\n","endScreenScore":2},"text":"One solution"},{"nextContentId":14,"feedback":{"title":"
Incorrect. <\/p>\n","subtitle":"
Revise the number of possible solutions assoicated with each type of geometric interpretation. <\/p>\n","endScreenScore":-1},"text":"Multiple solutions"},{"nextContentId":14,"feedback":{"title":"
Incorrect<\/p>\n","subtitle":"
Revise the number of possible solutions assoicated with each type of geometric interpretation. <\/p>\n","endScreenScore":-1},"text":"No solutions"}],"question":"
Consider the following system of equations:<\/h3>\n\n
2x<\/em> - y<\/em> + z<\/em> = 5<\/p>\n\n
x<\/em> + y<\/em> + 2z<\/em> = 7<\/p>\n\n
3x<\/em> + 4y<\/em> - z<\/em> = 10<\/p>\n\n
How many solutions will this system have?<\/p>\n"}},"subContentId":"497c0284-7d62-43d1-a0ff-8143e1abd6d4","metadata":{"contentType":"Branching Question","license":"U","title":"Q3a"}},"showContentTitle":false,"forceContentFinished":"useBehavioural","feedback":{"subtitle":""},"contentBehaviour":"useBehavioural"},{"type":{"library":"H5P.BranchingQuestion 1.0","params":{"branchingQuestion":{"alternatives":[{"nextContentId":15,"feedback":{"title":"
Oh dear. That's not right.<\/p>\n","subtitle":"
Revise the number of possible solutions assoicated with each type of geometric interpretation. <\/p>\n","endScreenScore":-1},"text":"One solution"},{"nextContentId":13,"feedback":{"title":"
Good one!<\/p>\n","subtitle":"
That's right. Systems that have consistent but dependent equations will have multiple solutions because the planes intersect at more than one point. For example, at a common line or coincient planes.<\/p>\n","endScreenScore":2},"text":"Multiple solutions"},{"nextContentId":15,"feedback":{"title":"
Incorrect<\/p>\n","subtitle":"
Revise the number of possible solutions assoicated with each type of geometric interpretation. <\/p>\n","endScreenScore":-1},"text":"No solutions"}],"question":"
Consider the following system of equations:<\/h3>\n\n
6x<\/em> + 3y<\/em> - 9z<\/em> = 18<\/p>\n\n
2x<\/em> + y<\/em> - 3z<\/em> = 6<\/p>\n\n
-4x<\/em> - 2y<\/em> + 6z<\/em> = -12<\/p>\n\n
How many solutions will this system have?<\/p>\n"}},"metadata":{"contentType":"Branching Question","license":"U","title":"Q3b"},"subContentId":"560aedf7-0b26-4706-9064-d05e45c6de6b"},"showContentTitle":false,"forceContentFinished":"useBehavioural","feedback":{"subtitle":""},"contentBehaviour":"useBehavioural"},{"type":{"library":"H5P.BranchingQuestion 1.0","params":{"branchingQuestion":{"alternatives":[{"nextContentId":-1,"feedback":{"title":"
Incorrect.<\/p>\n","subtitle":"
It looks like you find interpreting systems of equations challening. Contact your teacher to let them know you need some help.<\/p>\n","endScreenScore":2},"text":"One solution"},{"nextContentId":-1,"feedback":{"title":"
Incorrect. <\/p>\n","subtitle":"
It looks like you find interpreting systems of equations challening. Contact your teacher to let them know you need some help.<\/p>\n","endScreenScore":-1},"text":"Multiple solutions"},{"nextContentId":-1,"feedback":{"title":"","subtitle":""},"text":"No solutions"}],"question":"
Consider the following system of equations:<\/h3>\n\n
2x<\/em> - 2y<\/em> + 3z<\/em> = 5<\/p>\n\n
-2x<\/em> + 2y<\/em> - 3z<\/em> = 7<\/p>\n\n
4x<\/em> - 4y<\/em> + 6z<\/em> = 17<\/p>\n\n
How many solutions will this system have?<\/p>\n"}},"metadata":{"contentType":"Branching Question","license":"U","title":"Q3c"},"subContentId":"a35f2f6a-e4e7-42ad-880b-94dcbd75c00e"},"showContentTitle":false,"forceContentFinished":"useBehavioural","feedback":{"subtitle":""},"contentBehaviour":"useBehavioural"},{"type":{"library":"H5P.AdvancedText 1.1","params":{"text":"
Determining the number of possible solutions<\/h3>\n\n
When determining the number of solutions for a system of equations, consider the geometric interpretation. Every point at which all three planes intersect is a solution. So, consider the number of points at which all three planes intersect.<\/p>\n\n
OK. Let's try looking at a different system.<\/p>\n"},"metadata":{"contentType":"Text","license":"U","title":"Numbers of solutions"},"subContentId":"2960a8ca-624e-49ed-b402-8f80a2a55324"},"showContentTitle":false,"forceContentFinished":"useBehavioural","feedback":{"subtitle":""},"contentBehaviour":"useBehavioural","nextContentId":12},{"type":{"library":"H5P.AdvancedText 1.1","params":{"text":"
Determining the number of possible solutions<\/h3>\n\n
When determining the number of solutions for a system of equations, consider the geometric interpretation. Every point at which all three planes intersect is a solution. So, consider the number of points at which all three planes intersect.<\/p>\n\n
\n\t
Where the planes intersect at a single point, there is only one, unique solution for the system.<\/li>\n\t
Where planes intersect along a common line or overlap completely (are coincident), there are many possible solutions.<\/li>\n\t
When not all three planes intersect at the same place, such as two parallel planes and an intersecting third plane, there can be no solutions. <\/li>\n<\/ul>\n\n
OK. Let's try one more time.<\/p>\n"},"metadata":{"contentType":"Text","license":"U","title":"Numbers of solutions"},"subContentId":"1683f24b-349c-4299-bb69-9b4eb59540f5"},"showContentTitle":false,"forceContentFinished":"useBehavioural","feedback":{"subtitle":""},"contentBehaviour":"useBehavioural","nextContentId":13}],"endScreens":[{"endScreenTitle":"
Nicely done!<\/p>\n","endScreenSubtitle":"
It looks like you have a good understanding of how to interpret the geometry of 3x3 systems of equations. <\/p>\n\n
Now, let's go see how to solve them!<\/p>\n","contentId":-1,"endScreenScore":0,"endScreenImage":{"path":"images\/endScreenImage-64d47ff3e843d.jpg","mime":"image\/jpeg","copyright":{"license":"U"},"width":5184,"height":3456}}],"scoringOptionGroup":{"scoringOption":"dynamic-score","includeInteractionsScores":true},"startScreen":{"startScreenTitle":"
Understanding Geometric Interpretations of 3x3 Systems of Equations<\/p>\n","startScreenSubtitle":"
In this activity, test your understanding of the steps involved in interpreting the geometry of 3x3 systems of simultaneous equations.<\/p>\n"},"behaviour":{"enableBackwardsNavigation":false,"forceContentFinished":false},"l10n":{"startScreenButtonText":"Start the course","endScreenButtonText":"Restart the course","backButtonText":"Back","proceedButtonText":"Proceed","disableProceedButtonText":"Require to complete the current module","replayButtonText":"Replay the video","scoreText":"Your score:","fullscreenAria":"Fullscreen"}}}

node-9850_3x3 Branching scenario