Question 1

Explanation

Let the first number be $A$ and the second number be $B$.

From the problem, we get the following system of equations:

1. Twice the first number minus three times the second number equals $14$:

   $$2A - 3B = 14$$

2. The first number multiplied by $k$ plus twice the second number equals $4$:

   $$Ak + 2B = 4$$

3. The value of $k$ is equal to the sum of the first and second numbers:

   $$k = A + B$$

   With the condition $k > 0$.

Solution Steps

First, substitute equation $(3)$ into equation $(2)$:

Factor the terms containing $B$:

Note that $4 - A^2$ is a difference of squares, so it can be factored as $(2 - A)(2 + A)$:

Since $2 + A$ is the same as $A + 2$, we can divide both sides by $A + 2$ (assuming $A \neq -2$):

Next, substitute the value of $B$ into equation $(1)$:

So, the value of the first number is $4$.

Verification

We can verify the result:

If $A = 4$, then $B = 2 - 4 = -2$.

The value of $k = A + B = 4 + (-2) = 2$.

The condition $k > 0$ is satisfied since $2 > 0$.

Check equation $(1)$: $2(4) - 3(-2) = 8 + 6 = 14$ (Correct).

Check equation $(2)$: $4(2) + 2(-2) = 8 - 4 = 4$ (Correct).