For a relation 𝑅 from 𝑋 to 𝑌, in which set 𝑋 contains the numbers six, eight, 10, 13, 18 and set 𝑌 contains the numbers 10, 14, 18, 19, 21, 22, 24, 𝑎 𝑅𝑏 means 𝑏 is equal to two 𝑎 minus two. Find the value of 𝑥 if the ordered pair 𝑥, 14 is an element of 𝑅, where 𝑎 is an element of set 𝑋 and 𝑏 is an element of 𝑌.
We recall that a relation is a set of ordered pairs 𝑎, 𝑏 which follow a certain rule. In this case, we are told that 𝑏 is equal to two 𝑎 minus two. We are also told that the ordered pair 𝑥, 14 is an element of 𝑅. We can therefore substitute 𝑥 for 𝑎 and 14 for 𝑏 into our equation. 14 is equal to two multiplied by 𝑥 minus two. This can be rewritten as 14 is equal to two 𝑥 minus two. Adding two to both sides of this equation gives us two 𝑥 is equal to 16. We can then divide both sides of this equation by two such that 𝑥 is equal to eight.
We can see that eight is an element of set 𝑋 and 14 is an element of set 𝑌. This means that the correct answer, the value of 𝑥, is eight. One of the ordered pairs in the relation 𝑅 is eight, 14. The ordered pair six, 10 is also contained in the relation 𝑅 as two multiplied by six minus two is equal to 10. We also have the ordered pair 10, 18 as this also follows the rule. Finally, we have the ordered pair 13, 24 as two multiplied by 13 minus two is equal to 24. The relation 𝑅 is the set of four ordered pairs: eight, 14; six, 10; 10, 18; and 13, 24.