### Video Transcript

Determine the value of ๐ that makes the system of equations seven ๐ฅ plus five ๐ฆ equals seven and 35๐ฅ plus 25๐ฆ equals ๐ have infinitely many solutions.

Well, when weโre looking at system equations, there are three possible outcomes. The first outcome is the outcome where there are no solutions. And this occurs if the two equations conflict. For example, if we had ๐ plus ๐ equals seven and ๐ equals ๐ plus seven, these would be conflicting. And thatโs cause if we rearrange the second equation, we get ๐ minus ๐ equals seven. So therefore, weโd have two conflicting equations because weโd have ๐ plus ๐ is seven and also ๐ minus ๐ is seven. So therefore, thereโll be no solutions.

The second outcome is one that gives us infinite solutions. And itโs the one that weโre looking for in this question. And that is, if one equation can be deduced from another, then as we said thereโll be infinite solutions. And weโll look at that in more detail when we try and solve this problem.

And the final outcome is where thereโs a unique solution. And this occurs if the two equations cannot be deduced from one another and there are no conflicts. So as we said, itโs the second outcome that weโre looking at because what we want to do is find out what value of ๐ is gonna give us infinite solutions for our system of equations. And for that to occur, what we need to do is deduce one of our equations from the other one.

Well, if we look at the first two terms, we can see that if we multiply the first equation by five, then we will get the second equation, because seven multiplied by five is 35 and five multiplied by five is 25. So we get our 35๐ฅ plus 25๐ฆ. So therefore, we can deduce the second equation from the first equation. But in order to make this true, theyโll also have to have the same result. So itโd mean that weโd have to multiply the seven by five as well. So therefore, we can say that ๐ is gonna be equal to seven multiplied by five. So weโre gonna say that we get ๐ is equal to 35.

So therefore, we can say that if we want to make our system of equations have infinitely many solutions and we want to, therefore, deduce the second equation from the first equation. Then what we need to do is multiply seven by five to find the value of ๐. And as we said, that gives us a value of ๐ of 35.