How to solve for two variables $10x + 20y \geq 10203$

Question

I have a few equations of the form

$$10x + 20y \geq 10203$$

Basically just two variables set equal to a value. Each equation is unrelated. So basically the goal is to find an x and y that is equal to or greater than the final value on the right, with the ideal being so it's equal to the one on the right (so as close as possible to it). I don't know if I ever learned how to solve for more than one variable, so wondering how to do this.

If there is no solution, then any equation of similar form will do. Would be helpful to show the work because I don't understand how this can be solved without trial and error.

Answer 1

The locus (solution space) defined by an equation of the form $$ax+by+c=0$$ is an affine line in the plane.

If you change $=$ to $\geq$ or $\leq$ this defines the half-plane above the line or the half-plane below the line respectively. Try depicting these using CAS software like geogebra.

Answer 2

By using Bézout's identity we can say that $$10x+20y=\gcd{(10,20)}=10$$ is the smallest integer value that the summation of $10x+20y$ can take. So we know that it is possible for $$10x+20y=10200$$ and $$10x+20y=10210$$ for suitable $x,y$ but there can be no other value taken by $10x+20y$ between these values. i.e. the smallest solution to the problem is $$10x+20y=10210$$ where $x=-1021$ and $y=1021$.