Chemical Equations Chemical change involves a reorganization of the atoms in one or more substances.
System of Equations 2 (or more) equations, each of which has 2 (or more) variables.
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Transcript of System of Equations 2 (or more) equations, each of which has 2 (or more) variables.
![Page 1: System of Equations 2 (or more) equations, each of which has 2 (or more) variables.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649f295503460f94c43172/html5/thumbnails/1.jpg)
![Page 2: System of Equations 2 (or more) equations, each of which has 2 (or more) variables.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649f295503460f94c43172/html5/thumbnails/2.jpg)
System of Equations
2 (or more) equations, each of which has 2 (or more) variables
![Page 3: System of Equations 2 (or more) equations, each of which has 2 (or more) variables.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649f295503460f94c43172/html5/thumbnails/3.jpg)
![Page 4: System of Equations 2 (or more) equations, each of which has 2 (or more) variables.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649f295503460f94c43172/html5/thumbnails/4.jpg)
Our goal is normally to find a the values of the variables that work in both equations.
![Page 5: System of Equations 2 (or more) equations, each of which has 2 (or more) variables.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649f295503460f94c43172/html5/thumbnails/5.jpg)
In most cases you can think of this as the point where two lines cross.
(x,y) is thesolution tothe systemof equations.
![Page 6: System of Equations 2 (or more) equations, each of which has 2 (or more) variables.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649f295503460f94c43172/html5/thumbnails/6.jpg)
There are lots of ways to solve systems of equations.
One way is by graphing. Carefully graph both lines
on the same axes. Find the point where they
cross.
![Page 7: System of Equations 2 (or more) equations, each of which has 2 (or more) variables.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649f295503460f94c43172/html5/thumbnails/7.jpg)
Solve y = 2x + 2y = x – 1
![Page 8: System of Equations 2 (or more) equations, each of which has 2 (or more) variables.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649f295503460f94c43172/html5/thumbnails/8.jpg)
Solve y = 2x + 2y = x – 1
y = 2x + 2 y-intercept = 2 slope = 2/1
y = x – 1 y-intercept = -1 slope = 1/1
![Page 9: System of Equations 2 (or more) equations, each of which has 2 (or more) variables.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649f295503460f94c43172/html5/thumbnails/9.jpg)
The solution is(3, -4)
![Page 10: System of Equations 2 (or more) equations, each of which has 2 (or more) variables.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649f295503460f94c43172/html5/thumbnails/10.jpg)
Solve y = x + 2y = -x + 4
y = x + 2 y-intercept = 2 slope = 1/1
y = -x + 4 y-intercept = 4 slope = -1/1
![Page 11: System of Equations 2 (or more) equations, each of which has 2 (or more) variables.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649f295503460f94c43172/html5/thumbnails/11.jpg)
The solution is(1,3)
![Page 12: System of Equations 2 (or more) equations, each of which has 2 (or more) variables.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649f295503460f94c43172/html5/thumbnails/12.jpg)
Solve x + 2y = 52x + y = 4
![Page 13: System of Equations 2 (or more) equations, each of which has 2 (or more) variables.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649f295503460f94c43172/html5/thumbnails/13.jpg)
Solve x + 2y = 52x + y = 4
Find the intercepts.
![Page 14: System of Equations 2 (or more) equations, each of which has 2 (or more) variables.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649f295503460f94c43172/html5/thumbnails/14.jpg)
Solve x + 2y = 52x + y = 4
Find the intercepts.x + 2y = 5
(0,2.5) and (5,0)2x + y = 4
(0,4) and (2,0)
![Page 15: System of Equations 2 (or more) equations, each of which has 2 (or more) variables.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649f295503460f94c43172/html5/thumbnails/15.jpg)
Thesolutionis(1,2)
![Page 16: System of Equations 2 (or more) equations, each of which has 2 (or more) variables.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649f295503460f94c43172/html5/thumbnails/16.jpg)
Solve 2x + 2y = 64x – 6y = 12
![Page 17: System of Equations 2 (or more) equations, each of which has 2 (or more) variables.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649f295503460f94c43172/html5/thumbnails/17.jpg)
Solve 2x + 2y = 64x – 6y = 12
Intercepts2x + 2y = 6
(0,3) and (3,0)4x – 6y = 12
(0,-2) and (3,0)
![Page 18: System of Equations 2 (or more) equations, each of which has 2 (or more) variables.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649f295503460f94c43172/html5/thumbnails/18.jpg)
We can tell without even finishing the graph that (3,0) is the solution.
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Solve 3x + 2y = 123x + 2y = 6
![Page 20: System of Equations 2 (or more) equations, each of which has 2 (or more) variables.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649f295503460f94c43172/html5/thumbnails/20.jpg)
Solve 3x + 2y = 123x + 2y = 6
Intercepts3x + 2y = 12
(0,6) and (4,0)3x + 2y = 6
(0,3) and (2,0)
![Page 21: System of Equations 2 (or more) equations, each of which has 2 (or more) variables.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649f295503460f94c43172/html5/thumbnails/21.jpg)
These areparallel lines.
They both have
a slope or -3/2.
They never intersect, so there is no solution.
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It’s also possible to have infinitely many solutions, which would mean you have different expressions for the same line.
For example: 5x + 2y = 10y = -5/2x + 5
![Page 23: System of Equations 2 (or more) equations, each of which has 2 (or more) variables.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649f295503460f94c43172/html5/thumbnails/23.jpg)
There are lots of word problems that involve systems of equations. For instance …
Two tacos and a burrito cost $8. One taco and two burritos cost $10. How much is a taco, and how much is a burrito?
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Two tacos and a burrito cost $8. One taco and two burritos cost $10. How much is a taco, and how much is a burrito?
2t + 1b = 8 2x + 1y = 81t + 2b = 10 1x + 2y = 10
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2x + 1y = 81x + 2y = 10
The solution is (2,4)
![Page 26: System of Equations 2 (or more) equations, each of which has 2 (or more) variables.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649f295503460f94c43172/html5/thumbnails/26.jpg)
Two tacos and a burrito cost $8. One taco and two burritos cost $10. How much is a taco, and how much is a burrito?
(2,4) means a taco costs $2 and a burrito costs $4.