Help plz with this
There are 97 students in South High School. South High School offers only Chinese and Spanish. There are more students in Chinese than in Spanish, and every student takes at least one language. If students take only Spanish, then how many take both languages?
Let's assume that there are x students taking Spanish and y students taking Chinese. Since there are 97 students in total and every student takes at least one language, we know that:
x + y = 97
We also know that there are more students in Chinese than in Spanish, so:
y > x
Now, let's consider the scenario where only x students take Spanish. In this case, all the remaining students (97 - x) must be taking Chinese. But we also know that every student takes at least one language, so:
97 - x ≥ 1 x ≤ 96
Combining these inequalities, we get:
1 ≤ x ≤ 96
Now, we need to find the number of students who take both languages. Let's call this number z. Then, we have:
z = y - x
We want to find the value of z when all the students taking Spanish are also taking Chinese. In other words, we want to find the value of z when x = z.
Substituting x = z into the first equation (x + y = 97), we get:
z + y = 97
Substituting x = z into the second equation (y > x), we get:
y > z
Now we have a system of two equations:
z + y = 97 y > z
We want to find the value of z that satisfies both equations. One way to do this is to try different values of z and see which ones work. However, we can also use algebra to solve the system.
Subtracting z from both sides of the first equation, we get:
y = 97 - z
Substituting this expression for y into the second equation (y > z), we get:
97 - z > z
Simplifying this inequality, we get:
97 > 2z
Dividing both sides by 2, we get:
48.5 > z
Since z is an integer, the only value that satisfies both inequalities is:
z = 48
Therefore, when all the students taking Spanish are also taking Chinese, there are 48 students taking both languages.