Use the substitution or elimination method to solve each syste...

Question

Use the substitution or elimination method to solve each system of equations. Identify any inconsistent systems or systems with infinitely many solutions. If a system has infinitely many solutions, write the solution set with y arbitrary.
$\frac{1}{6 x} + \frac{1}{3 y} = 8$
$\frac{1}{4 x} + \frac{1}{2 y} = 12$

$\frac{1}{6 x} + \frac{1}{3 y} = 8$

$\frac{1}{4 x} + \frac{1}{2 y} = 12$

Accepted Answer

Hello everybody. I'm today today. We're going to look at this math question that states solve the given system of linear equations using the elimination method. Now, the system equations includes X divided by seven plus Y divided by 14 equals eight and X divided by four plus Y divided by eight equals 14. Now the answers provided are A X comma 112 minus two X B X comma 112 minus X C X comma 112 plus two X and D no solution. Now, when we think about the elimination method, we know that we want to try to eliminate one of the variables. So I see that we have in both equations, we have fractions. So I'm gonna try to get rid of those fractions. See what if we can work with that better. So I'm going to label X divided by seven plus Y divided by 14 equals eight. That'll be equation one and X divided by four plus Y divided by eight equals 14. That'll be equation two. So for equation one, I'll have equation one and I'll multiply that by 14 to get rid of the fractions. So once we do that. We'll have X multiplying 14. So we'll have 14, X divided by seven plus 14, Y divided by 14. And finally eight multiplying 14, 14, X divided by seven is two X. So I have two X plus and then 14, Y divided by 14 is just Y and eight multiplying 14 is 100 12. And that will be our new equation one. Now for equation two, I'll multiply that by eight. So I'll have eight X divided by four. Now plus A Y divided by A, it was eight multiplying the 14. So eight X divided by four is two X plus eight Y divided by eight, which is just Y and finally eight multiplying of which is 100 12. Now we see that both equations two X pr two X plus Y equals 1 12. So both equations are the same, meaning that the system of equations that we have has infinitely many solutions since they're both the same. So all we can do now is use one of the equations to solve for a why and see what that, that Y solution would look like. So in this case, I'm just going to use the same two X plus Y equals 12. And also, so for Y there, so we'll have Y is equal to 12 minus two X. And that's going to be our Why we can't solve it further because like I said, there will be many infinitely solutions. So we'll just have uh solutions of X comma 1 12 minus two X. There's no specific X here and no specific Y because there are infinite solutions. So in this case, this answer matches with answer choice A X comma 1 12 minus two X. So I hope that was helpful. And until next time.

Key Concepts

Systems of Linear Equations

Substitution and Elimination Methods

Types of Solutions: Consistent, Inconsistent, and Dependent Systems

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