# What are the points of intersection for #y = 2x + 3# and #y = x + 5#?

##### 1 Answer

Suppose we separated the variables into

#\mathbf(y_1 = 2x_1 + 3)#

#\mathbf(y_2 = x_2 + 5)#

The **point of intersection** occurs when the two graphs have **equal** values of *at the same time*. There is *only* **one solution**, because two straight lines can only intersect once. (On the other hand, two curved lines may intersect twice.)

The solution will be the **coordinate**

What we can do to proceed is assume that

#2x_1 + 3 = x_2 + 5#

#= x_1 + 5#

Subtract

#x_1 + 3 = 5#

Then I would subtract

#color(blue)(x_1 = x_2 = 2)#

Now, since the solution coordinate requires that we have *both*

#color(blue)(y_1) = 2x_1 + 3#

#= 2(2) + 3 = color(blue)(7)#

And just to show that indeed

#color(green)(y_2) = x_2 + 5#

#= x_1 + 5#

#= 2 + 5#

#= color(green)(7 = y_1)#

Finally, that means our solution coordinate is:

#color(blue)("("2,7")")#