y = x + 3
3x + y = -1

Graph the first line by getting some points:

For the first line, get three points.  Choose 3
arbitrary numbers for one letter, substitute them
into the first equation and solve for the other
letter. Them make them into a point (x,y)

Arbitrarily choose 1 to substitute for y

                   y = x + 3 
                   1 = x + 3

subtract 3 from both sides

                  -2 = x

plot the point (-2,1)

Arbitrarily choose 0 to substitute for x

                   y = x + 3 
                   y = 0 + 3
                   y = 3

plot the point (0,3)

Arbitrarily choose 2 to substitute for x

                   y = x + 3 
                   y = 2 + 3
                   y = 5

plot the point (2,5)

Draw a red line through these three points:



Likewise for the second line, get three points.

Arbitrarily choose 5 to substitute for y

                   3x + y = -1
                   3x + 5 = -1

subtract 5 from both sides

                       3x = -6

Divide both sides by 3

                        x = -2

plot the point (-2,5) on the same graph.

Arbitrarily choose 0 to substitute for x

                   3x + y = -1
                 3(0) + y = -1
                    0 + y = -1
                        y = -1
plot the point (0,-1)

Arbitrarily choose 1 to substitute for x

                   3x + y = -1
                 3(1) + y = -1
                    3 + y = -1
                    
subtract 3 from both sides

                        y = -4

plot the point (1,-4)

Draw a green line through these three points:

Now you have this graph:


Notice the the red and green lines cross at the point
which is directly above -1 on the x axis, and which
is directly to the left of 2 on the y-axis.

Therefore the solution is (x,y) = (-1,2) which
means x = -1 and y = 2

Now we check to see if the solution is correct.

We substitute both x = -1 and y = 2 into the first
equation:

                   y =  x + 3 
                   2 = -1 + 3
                   2 = 2

So the first equation checks.  But that is not
enough. We must also substitute both x = -1 and y = 2 
into the second equation as well.

                    3x + y = -1
                 3(-1) + 2 = -1
                    -3 + 2 = -1
                        -1 = -1
                 
That equation checks also, so (-1,2) is the correct
solution.

Edwin