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# Module 5

### solving system and linear equations

x+y=6 4x+3y=22 substitute A and B in equation a. (4,2) b. (2,4) | with the first equation being (a) And second equation being (b) just plug in..... 4+2=6 the answer is yes 4(2)+3(4)=22 8+12=22? that is a no |

Another example | *Hint* we are only substituting what we are giving in. follow order of operations!!!!! |

x+y=7 (2,5) 2x+3y=19 (4,3) | On the back of this do this yourself...Then I will go through it with you. |

Now substitute 2+5=7? 2(4)+3(3)=19? | Answer is yes no |

Now onto solve linear equations by graphing. | x+y=8 x-y=6 now you wanna graph your own numbers that will fit on a smaller graph. so try to stay within numbers of 10. |

x+y=8 lets try for (x) for being the input number 0,1,2 (y)will be the output number/or answer 0+y=8 y=8 (0,8) 1+y=8 minus 1 from both sides y=7 (1,7) 2+y=8 minus 2 both sides y=6 (2,6) | x-y=6 lets plug in 0,1,2 0-y=6 -y=6 cant be a negative y. So multiply both sides by -1 y=-6 (0,-6) 1-y=6 -y=5 y=-5 (1,-5) 2-y=6 -y=4 y=-4 (2,-4) now graph!!!! |

now with graphing equations you're now all done. you want to find the intersecting numbers. | once you graph points will either intersect which would be the correct points for the answer. or they will be parallel which has no answer/ no solution. or they land on the same path/same line so infinitely amount of numbers!!! |

Understand two lines that intercept they have one point in common. when lines that are parallel there is no solution since they will never intercept. And when a line sits on another they have infinity amount of numbers!!! | So far we have talked about substituting what we are giving into the equations and picking our own numbers to graph to find out if they are intercepting or parallel or infinity amount of numbers. |

Next without graphing decide if the point are parallel, intercept, or identical lines. and how many solutions they have? | *remember* parallel has no solutions. intercepting has one solution. And identical lines have an infinity amount of solutions. |

4x+y=19 x+3y=3 Put in slope intercept form | What is Slope intercept form *y=mx+b* |

y=-4x+19 Slope is -4 3y=-x+3 divide 3y by 3 to get rid of it. and divide by 3 on other side. y=-1/3x+1 Slope is -1/3 | since slope is not the same its one solution which means it intercepts. |

Now we will start substitution an equation into another equation. Sounds weird but its not to hard I promise. | x+y=4 x=3y its simply since x is solved for |

3y+y=4 (3y) is x=3y combine like terms 4y=4 divide by 4 y=1 | Now substitute y=1 into one of the equations x=3(1) x=3 now put both x and y into the equation to check 3+1=4 true (3,1) |

Staying with substitution method 8x+2y=4 -4x=y+8 | |

One more substitution 1/5x-y=2 x-5y=10 equation number two minus -5y both sides x=5y+10 sub into equation number 1 | 1/5(5y+10)-y=2 y+2-y=2 combine like terms 2=2 means infinitely many solutions |