Let

x = the $ invested at 4%
y = the $ invested at 7%

The financial board of a corporation invested $25,000 means:

x + y = 25000

The first-year combined income from the two investments was $1300:

0.04*x + 0.07*y = 1300

by solving the system of equations

x + y = 25000
0.04*x + 0.07*y = 1300

we find

x = $15,000

y = $10,000

click here to see the step by step solution of the system of equations

$15,000 we invested at 4% and $10,000 were invested at 7% annual simple interest rate.