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I don't know where they got the 6 from. Clearly 6 does not equal 3. I'm not sure what they're saying there.

To find the intersection of the lines, you can do the following:

At the point of intersection, notice y = -3x + 3 and y = x-5.

Then -3x + 3 = x-5.

Rearranging, we get:

4x = 8

Which gives x = 2.

Then to find the value of y, we substitute our value of x into one of the original equations:

y = -3x + 3 = -3(2) + 3 = -3

So we have x = 2 and y = -3, or the coordinate (2, -3).
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It's not a 6. It is a letter b.

They are just putting each equation in slope intercept form, y = mx + b.

The first equation becomes y = x - 5. Comparing that to y = mx + b you can identify m = 1 and b = -5.

The second equation becomes y = -3x + 3. Comparing that to y = mx + b you can see that m = -3 and b = 3.

In each case "y-int = b = ..." means "the y intercept is b which is..."

They use this information to draw each line, and then they determine the point of (2, -3) directly from the graph. After drawing the lines carefully you can see that is the point of intersection.

You are probably being taught how to solve a system of equations graphically. If you are ever asked to do this, with the word GRAPHICALLY, this is what you are expected to do. Draw the lines for each equation, as carefully and accurately as possible, then read the point of intersection from the graph.

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