Question

If y varies directly with x and y = 25 when x = 5. How do you write the direct linear variable equation and what is the constant?

Answer 1

Hint: We are given y varies directly with x, to find the equation for this we will first see what type of relations we have and we will learn about inverse and direct relation and using these we will find the relation suitable for our problem. We will use that directly as the relation is given as y = kx where k is a constant. Then we will use x = 5, y = 25 to find the value of k.

Complete step-by-step solution:
We are given that y varies directly with x, we are asked to write the direct linear variation equation. Before we move on to that we must learn what direct relation means. In maths, there are two types of relation, one is called direct relation and the other is indirect relation (inversely). In direct relation, two things are connecting directly means they both will behave the same way if there is a decrement in one thing then the other will decrease too and if one is increasing then other will increase too, say we have two things as A and B. They are denoted as \[A\propto B.\] The sign \['\propto '\] is called a sign of proportionality. To form an equation we need to change the proportional \[\left( \propto \right)\] to equal (=). So, we will multiply by constant. So, the equation becomes A = kB. The other type of relation is indirectly where A is moving opposite of B. If A increases, B will decrease and if A decreases B will increase. It is represented as \[A\propto \dfrac{1}{B}.\]
Now, in our problem, we have that we have two things as y and x. We also have that y is directly proportional to x. So, as we learn above, we will get, \[y\propto x.\] To change \[\propto \] to =, we will multiply by constant, say k, we will get \[y=kx.\] So, the required equation is y = kx. Now, as we have for x = 5, we have y = 25, so putting them in our equation, we get,
\[25=k\left( 5 \right)\]
Dividing both the sides by 5, we get,
\[\Rightarrow 5=k\]
So, the value of the constant k is 5.
Hence the equation is y = 5x.

Note: The graph of the direct relation will always produce a straight line. If the constant of proportionality is positive, then the graph will have a positive gradient. If the constant is negative, then the graph will have a negative gradient.