Let's represent the lines $2x + 3y = t$ and $5x - 7y = t$ in a Cartesian coordinate system, where $t$ is a real number. What will be the geometric locus of the intersection points of these lines as $t$ takes on all possible values?
If a point $P(x, y)$ is the intersection point of the lines

$$
2 x+3 y=t
$$

$$
5 x-7 y=t
$$

for some $t$, then both (1) and (2) must hold for its coordinates. This means that if we substitute the coordinates of $P$ into the left-hand side of either (1) or (2), we get the same number, so for the coordinates of $P$,

$$
2 x+3 y=5 x-7 y
$$

Conversely, if the coordinates of some point $P(x, y)$ satisfy (3), then by choosing the common value of the two sides of (3) as $t$, the equations (1) and (2) also hold for them. Therefore, a point $P$ belongs to the geometric locus in question if and only if its coordinates satisfy (3). Since this is the equation of a line, the geometric locus is a line. It can be seen from the equation that it passes through the origin, and its slope is 0.3.