Question 1168173
Let $i$ be Lanre's original income from interest, and $w$ be his original income from wages.

From the first statement, his total original income is $4000:
Equation 1: $i + w = 4000$

From the second statement, he doubles his investment (which we assume directly doubles his interest income) and his total income becomes $6500. His wage income remains the same.
New interest income = $2i$
New wage income = $w$
New total income = $6500$
Equation 2: $2i + w = 6500$

We have a system of two linear equations with two variables:
1) $i + w = 4000$
2) $2i + w = 6500$

We will use Cramer's rule to solve this system. Cramer's rule states that for a system of linear equations:
$a_1x + b_1y = c_1$
$a_2x + b_2y = c_2$

The solutions for $x$ and $y$ are given by:
$x = \frac{D_x}{D}$
$y = \frac{D_y}{D}$

where $D$ is the determinant of the coefficient matrix, and $D_x$ and $D_y$ are the determinants of matrices formed by replacing the column corresponding to $x$ and $y$ respectively with the constant terms.

In our case, the variables are $i$ and $w$:
$1i + 1w = 4000$
$2i + 1w = 6500$

Here, $a_1 = 1, b_1 = 1, c_1 = 4000$ and $a_2 = 2, b_2 = 1, c_2 = 6500$.

First, calculate the determinant of the coefficient matrix $D$:
$D = \begin{vmatrix} 1 & 1 \\ 2 & 1 \end{vmatrix} = (1 \times 1) - (1 \times 2) = 1 - 2 = -1$

Next, calculate the determinant $D_i$ by replacing the first column (coefficients of $i$) with the constant terms:
$D_i = \begin{vmatrix} 4000 & 1 \\ 6500 & 1 \end{vmatrix} = (4000 \times 1) - (1 \times 6500) = 4000 - 6500 = -2500$

Now, calculate the determinant $D_w$ by replacing the second column (coefficients of $w$) with the constant terms:
$D_w = \begin{vmatrix} 1 & 4000 \\ 2 & 6500 \end{vmatrix} = (1 \times 6500) - (4000 \times 2) = 6500 - 8000 = -1500$

Finally, use Cramer's rule to find the values of $i$ and $w$:
$i = \frac{D_i}{D} = \frac{-2500}{-1} = 2500$
$w = \frac{D_w}{D} = \frac{-1500}{-1} = 1500$

So, Lanre's original income from interest was $2500, and his original income from wages was $1500.

Let's check if these values satisfy the given conditions:
Original total income: $2500 + 1500 = 4000$ (Correct)
Doubled investment (interest) and increased total income: $2(2500) + 1500 = 5000 + 1500 = 6500$ (Correct)

Final Answer: The final answer is $\boxed{Original interest income was $2500 and original wage income was $1500}$