If the simultaneous equations six 𝑥 plus seven 𝑦 equals two and 18𝑥 plus 𝑘𝑦 equals six have an infinite number of solutions, what is the value of 𝑘?
Equations that have an infinite number of solutions are the same line. We’re told that these two equations have infinite number of solutions, and therefore they must be the same line. Let’s look carefully at how these two equations compare.
How would we get from six 𝑥 to 18𝑥? You multiply by three. The same is true for going from two to six. Since these equations are the same line and they would be graphed in the same way, we can multiply seven by three to find the value of 𝑘. Here, 𝑘 would equal 21.
If 𝑘 equals 21, then these two lines are equivalent, and we would graph them as the same line on a graph.