Question 704258
Parallel lines have the same slope,
because slope is the change in y for an x increase of 1,
and that is the tangent of the angle the line makes with the positive x axis.
{{{drawing(300,270,0.1,1.1,-0.1,0.8,
grid(0),
green(line(0,-7/40,1.2,7/8)),
blue(triangle(0.2,0,1,00,1,0.7)),
blue(rectangle(0.95,0,1,0.05)),
line(0.2,0,0.2,-0.03),line(1,0,1,-0.03),
locate(0.17,0,a),locate(0.94,0,"a+"),
locate(0.6,0.07,1),locate(0.87,0.37,slope)
)}}}
 
{{{y=x+5}}} is the equation of a line in {{{highlight(slope-intercept)}}} form.
That is the unique and very useful form that starts with "y=".
The number added at the end (5 in this case) is the intercept,
the y value for x=0, and therefore the y- coordinate of the point where the line "intercepts" the y-axis.
The number multiplying the x (an invisible 1 in this case) is the slope.
You can see that for x=0, y=5, but if you increase x by one,
for x=1, y=1+5 has increased by 1.
 
The line parallel to {{{y=x+5}}} will have {{{highlight(slope=1)}}} and an intercept, {{{b}}} to be determined.
Its equation will be {{{y=x+b}}}
We know that for that line the 
point with {{{x=2}}} has {{{y=3}}} so substituting into {{{y=x+b}}} we get
{{{3=2+b}}} --> {{{3-2=b}}} --> {{{highlight(b=1)}}}
and the equation is {{{highlight(y=x+1)}}}