Lets say we have two credit cards. Card $A$ has a balance of \$2000 and card $B$ \$3500. That is, $C_A = 2000, C_B = 3500$. The interest rates are $r_A = 0.20, r_B = 0.25$. What is the optimal way to pay the balance to reduce interest if you have a budget of \$1500 for a single month?
Here is my work:
Since we are only accruing interest over one month, we get the following equations: $$\text{Total Card A }=(2000-x_1)\left(1+\frac{0.2}{12}\right)\\\ \text{Total Card B }=(3500-x_2)\left(1+\frac{0.25}{12}\right) $$ where $x_1$ and $x_2$ are the payments. We also know: $$x_1+x_2=1500$$
I'm not sure how to solve this since the two credit card equations are not equal to anything. I tried graphing them but none of these lines intersect with each other.
Card B: $(3500 - (1500 - x_1))\left(\frac{12.25}{12} \right)$
Total debt remaining = Card A $+$ Card B = $y$
$y = (2000 - x_1) \left(\frac{12.2}{12} \right) + (2000 + x_1) \left(\frac{12.25}{12} \right)$
Point of interest $(x, y) = (0, 4075)$
$y = (500 + x_2) \left(\frac{12.2}{12} \right) + (3500 - x_2) \left(\frac{12.25}{12}\right)$
Point of interest $(x, y) = (0, 4081.25)$
The point $(a, b)$ where the two lines above intersect is where $x_1 = x_2$ and the corresponding $y$ values (total debt) are equal. This solution is required if you want to split $1500$ to both credit cards, without affecting your debt. However what happens if the point of intersection is (say) $(3000, 4000)$? You only have $1500$ and $1500 < 3000$.