Question 1

We will use the given information to form two linear equations.

Step 1 — Write the equations

Let's use the given relation $y = ax + b$. The first observation is $x = \mathbf{10}$ and $y = \mathbf{400}$. We substitute these values into the relation.

$400 = a(10) + b$

$10a + b = 400 \quad \text{(Equation 1)}$

The second observation is $x = \mathbf{14}$ and $y = \mathbf{500}$. We substitute these values into the relation.

$500 = a(14) + b$

$\boxed{14a + b = 500 \quad \text{(Equation 2)}}$

Step 2 — Find the value of 'a'

We have two linear equations. Let's subtract Equation 1 from Equation 2.

$(14a + b) - (10a + b) = 500 - 400$

$14a - 10a + b - b = 100$

$4a = 100$

$a = \frac{100}{4}$

$\boxed{a = 25}$

Step 3 — Find the value of 'b'

Now we substitute the value of $a = \mathbf{25}$. Let's use Equation 1 for this.

$10a + b = 400$

$10(25) + b = 400$

$250 + b = 400$

$b = 400 - 250$

$\boxed{b = 150}$

Answer

(i) The value of a is 25. (ii) The value of b is 150.