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.x/2+ y/3 = 43x/2+3y/2 = 15

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 X divided by four plus Y divided by five is equal to nine. And the second given equation is five, X divided by four plus seven, Y divided by four is equal to 60. Here we have four answer choice options, answer choice A X and Y are equal to 20. Answer B X is equal to 15 and Y is equal to 25. Answer C X is equal to 30 and Y is equal to and answer D X is equal to 50 and Y is equal to negative 10. So utilizing the elimination method, we first need to select either of the variables of which we want to solve for. So we can first focus on solving for Y. And that means we need to manipulate both equations such that we get the same X terms. So we can start by taking the first equation and multiplying the left side and the right side by 20. So doing this, we get a new manipulated equation of five X plus four. Y is equal to 180. And now, for the second equation again, we want the same X terms. So we can multiply it the left side and the right side by four. And so multiplying by four, our second equation becomes five X plus seven Y is equal to 240. And so now to actually utilize the elimination method, we just need to subtract the second equation from the first equation. And so in doing this, we can begin with our X terms. So we have five X minus five X which gives us zero, then we have four Y minus seven Y which gives us minus three Y and then this is equal to the right side. And now subtracting these constants, we have 180 minus 240 which gives us negative 60. So now we have obtained a new equation of negative three, Y is equal to negative 60. Now just solving for Y, we need to divide both sides by negative three and we get Y is equal to positive 20. So now we have already found our value for Y. So we next need to focus on solving for X. And here we can utilize the two manipulated equations that we obtained from earlier. So just recalling, we had our first equation as five, X plus four, Y is equal to 180. And our second equation was five X plus seven Y is equal to 240. And so now we want to get our Y terms to be the same. And so, and so we can solve for X. So that means we first just want to multiply the left side and the right side of our first equation by 20 or by seven. And so multiplying both sides by seven, we get a new equation of 35 X plus 28 Y is equal to 1260. And now for the second equation, we can multiply the left side and the right side by either positive or negative four. But here, let's choose negative four. And so again, multiplying by negative four, we get a new equation of negative 20 X minus 28 Y is equal to negative 960. So here we see that our Y terms are the same coefficient value but opposite signs. So instead of subtracting these equations like we did previously, we now just want to add them together. So adding together, starting with our X terms, we have 35 X minus or plus negative X which gives us 15 X, we then have 28 Y plus negative 28 Y which gives us zero. Then on the right side, we have 1000, 260 plus negative 960 this gives us positive 300. So now we have obtained a new equation of 15 X is equal to 300. So now just solving for X, we just need to divide both sides by 15 and we see our value for X comes out to positive 20. So here we have found that our value for X and Y is 20. So we are left with answer choice. A where again, X is equal to 20 and Y is equal to 20 as well. 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.x/2+ y/3 = 43x/2+3y/2 = 15

Key Concepts

Systems of Equations

Elimination Method

Clearing Denominators

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