Solve each system by elimination. In systems with fractions, firs...

Question

Solve each system by elimination. In systems with fractions, first clear denominators. See Example 2.6x + 7y + 2 = 07x - 6y - 26 = 0

Accepted Answer

Hey, everyone. Here we are asked to use the elimination method to solve the following system of equations where the first given equation is two X plus five, Y plus 11 is equal to zero. And the second equation is three, X minus three, Y plus 27 is equal to zero. And here we have four answer choice options. Answer choice A X is equal to zero, Y is equal to one answer. Choice B X is equal to negative eight and Y is equal to one answer C X is equal to negative three and Y is equal to 14 and answer D no solution. So here utilizing the elimination method, we first need to select one of the variables that we want to solve for. So we can begin by focusing on solving for Y. And so that means we need to manipulate both equations so that we get the same X terms. So doing this, we can first multiply the first given equation by negative three. So we multiply the left side and the right side of this equation by negative three. And this gives us a new equation of negative six X minus 15 Y minus 33 is equal to zero. And then with the second equation again, we want the same X terms regardless of the sign. So here we will just multiply the left side and the right side by two. And so now multiplying by two, we get a second equation of six X minus six Y plus 54 is equal to zero. And so now we see here that our X terms have the same coefficient but different signs. So that means we have to add these two equations together. And so adding these equations, we can go to term by term. So starting with our X terms, we have negative six X plus positive six X which gives us zero. Then we have negative 15 Y plus negative six Y giving us negative 21 Y, then we have negative 33 plus 54 which gives us plus 21 this is equal to zero since zero plus zero is just zero. So now we have the equation negative 21 Y plus 21 is equal to zero. So solving for Y, we can first just add 21 Y to both sides and we get 21 Y is equal to 21. Now just getting Y alone, we divide both sides by 21 we see that our value for Y is just one. So now we can move on to solving for X. So again, looking back at our original equations. We have two X plus five, Y plus 11 is equal to zero. In the second equation, we have three X minus three, Y plus 27 is equal to zero. So now here we need to solve for X, which means we want both of our Y terms to be the same. So we will begin by multiplying both sides of the first equation by positive three. And so multiplying by three, we get the equation six X plus 15, Y plus 33 is equal to zero. And now with the second equation, we want to multiply both sides by five. And so we get a second equation of 15 X minus 15, Y plus 135 is equal to zero. So now again, we see here that our Y terms are the same coefficient but with different signs. So we just want to add both equations together so that our Y terms cancel out. And so going term by term, we have six X plus 15 X which gives us X. Then we have 15 Y plus negative 15 Y which gives us plus zero like we wanted. And then we have 33 plus 135 which gives us plus 168. And then we have zero plus zero. So this expression is set equal to zero. And so now we have a new equation of 21 X plus 168. Is equal to zero. So now just solving for X, we can begin by just subtracting 168 from both sides. So we have 21 X is equal to negative 168. So now finishing solving for X, we can divide both sides by 21. And so we get that X here is equal to negative eight. So here we have found our values for X and Y. So we are left here with answer choice B where again, X is equal to negative eight and Y is equal to one. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

Solve each system by elimination. In systems with fractions, first clear denominators. See Example 2.6x + 7y + 2 = 07x - 6y - 26 = 0

Key Concepts

Systems of Equations

Elimination Method

Clearing Denominators

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