Determine the value of 𝑘 that makes the system of equations seven 𝑥 plus five 𝑦 equals seven and 35𝑥 plus 25𝑦 equals 𝑘 have infinitely many solutions.
Well, when we’re 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 𝑎 plus 𝑏 equals seven and 𝑎 equals 𝑏 plus seven, these would be conflicting. And that’s cause if we rearrange the second equation, we get 𝑎 minus 𝑏 equals seven. So therefore, we’d have two conflicting equations because we’d have 𝑎 plus 𝑏 is seven and also 𝑎 minus 𝑏 is seven. So therefore, there’ll be no solutions.
The second outcome is one that gives us infinite solutions. And it’s the one that we’re looking for in this question. And that is, if one equation can be deduced from another, then as we said there’ll be infinite solutions. And we’ll look at that in more detail when we try and solve this problem.
And the final outcome is where there’s 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, it’s the second outcome that we’re looking at because what we want to do is find out what value of 𝑘 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 35𝑥 plus 25𝑦. So therefore, we can deduce the second equation from the first equation. But in order to make this true, they’ll also have to have the same result. So it’d mean that we’d have to multiply the seven by five as well. So therefore, we can say that 𝑘 is gonna be equal to seven multiplied by five. So we’re gonna say that we get 𝑘 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 𝑘. And as we said, that gives us a value of 𝑘 of 35.