Answer:
5)
Line 1: [tex]y=-x+3[/tex]
Line 2: [tex]y=3x-3[/tex]
6)
Line 1: [tex]y=\frac{1}{2}x-2[/tex]
Line 2: [tex]y=-\frac{5}{2}x+6[/tex]
7)
See below
Step-by-step explanation:
To write the system of equations on the graph, you just need to write the 2 equations of the 2 lines.
5)
The first thing to do is find the slope of the lines. Pick one of them, and start by finding 2 points on that line that land perfectly on the grid.
It's pretty hard to read that top graph in the photo, but I think I see these ones marked:
Line 1: (0, 3) and (3, 0)
Line 2: (0, -3) and (2, 3)
Make sure I got them correct, because the equations will be wrong otherwise. Same for problem 6. To find the slope, divide the change in y by the change in x. Starting with line 1, y went from 3 to 0, so a change of -3. x went from 0 to 3, so a change of 3.
Divide the change in y by the change in x:
[tex]m=\frac{y2-y1}{x2-x1}\\\\m=\frac{0-3}{3-0}\\\\m=\frac{-3}{3}\\\\m=-1[/tex]
The slope of line 1 is -1. That means every time x increases by 1, y decreases by 1. You can see that on the line.
Now for line 2:
[tex]m=\frac{3--3}{2-0}\\\\m=\frac{6}{2}\\\\m=3[/tex]
When x increases by 1, y increases by 3.
The last thing to do is find the y-intercept of both lines. That's just where the line intersects the y-axis, at x = 0, and it's clearly visible on the graph so there isn't much to do.
Line 1: 3
Line 2: -3
Now, you can write the equations for these lines in slope intercept form:
[tex]y=mx+b[/tex]
where m is the slope and b is the y-intercept we found above.
Line 1: [tex]y=-x+3[/tex]
Line 2: [tex]y=3x-3[/tex]
In line 1, -1x can be simplified to just -x.
6)
I'll do the exact same process here, but with less explaining. Let me know if you have any questions.
Find 2 points:
Line 1: (0, -2) and (4, 0)
Line 2: (0, 6) and (4, -4)
Find the slope of line 1:
[tex]m=\frac{y2-y1}{x2-x1}\\\\m=\frac{0--2}{4-0}\\\\m=\frac{1}{2}[/tex]
and line 2:
[tex]m=\frac{-4-6}{4-0}\\\\m=\frac{-10}{4}\\\\m=-\frac{5}{2}[/tex]
Now, the y-intercepts:
Line 1: -2
Line 2: 6
Write the equations:
Line 1: [tex]y=\frac{1}{2}x-2[/tex]
Line 2: [tex]y=-\frac{5}{2}x+6[/tex]
7)
A system with 1 solution means 2 lines that intersect at 1 point. Problems 5 and 6 are both examples of this. A system with no solution means lines that never intersect, meaning that they have the same slope, they're parallel.
Example:
[tex]y=2x+1\\y=2x+2[/tex]
Those both have a slope of 2, and if you graph them, you'll see that they never intersect.
A system with infinitely many solutions means they intersect at every possible point. These would just be the same line.
Example:
[tex]y=2x+1\\y=2x+1[/tex]
That's what they are, graph them however you like.