Worksheet for graphing linear equations on coordinate planes.

I understand the task is to sketch the graph of each linear equation provided on the coordinate grids. To solve this, I need to analyze each equation and determine how to plot its line.

Let me start by looking at the first equation: 1) x + 4y = 0.

First, look closely:
- The image shows a worksheet titled "Linear Equations" with six coordinate grids.
- Each grid has an equation next to it that needs to be graphed.
- For equation 1 (x + 4y = 0), I need to find points that satisfy this equation to plot the line.
- I can convert this to slope-intercept form (y = mx + b) to make graphing easier.

Next, find information:
To graph x + 4y = 0, I'll rearrange it to y = mx + b form:
x + 4y = 0
4y = -x
y = -1/4x

This tells me the line passes through the origin (0,0) and has a slope of -1/4.

Then, review the findings:
The line for equation 1 will pass through (0,0) and for every 4 units I move right, I move 1 unit down (since the slope is -1/4).

Now let me examine the second equation: 2) x + y = -8.

First, look closely:
- Equation 2 is x + y = -8.
- I need to convert this to slope-intercept form as well.

Next, find information:
x + y = -8
y = -x - 8

This tells me the line has a slope of -1 and a y-intercept of -8.

Then, review the findings:
The line for equation 2 will cross the y-axis at (0,-8) and have a slope of -1 (for every 1 unit right, go 1 unit down).

Let me continue with equation 3: 3) 5y = 3x - 0.

First, look closely:
- Equation 3 is 5y = 3x - 0, which simplifies to 5y = 3x.
- This is already close to slope-intercept form.

Next, find information:
5y = 3x
y = (3/5)x

This tells me the line passes through the origin (0,0) and has a slope of 3/5.

Then, review the findings:
The line for equation 3 will pass through (0,0) and for every 5 units I move right, I move 3 units up.

Let me examine equation 4: 4) 6x = 3y + 3.

First, look closely:
- Equation 4 is 6x = 3y + 3.
- I need to solve for y to get slope-intercept form.

Next, find information:
6x = 3y + 3
6x - 3 = 3y
y = 2x - 1

This tells me the line has a slope of 2 and a y-intercept of -1.

Then, review the findings:
The line for equation 4 will cross the y-axis at (0,-1) and have a slope of 2 (for every 1 unit right, go 2 units up).

Now let me look at equation 5: 5) 3x = 18 - 7.

First, look closely:
- Equation 5 appears to be 3x = 18 - 7, which simplifies to 3x = 11.
- This is not a typical linear equation in two variables; it's actually a vertical line.

Next, find information:
3x = 11
x = 11/3
x ≈ 3.67

This tells me the line is vertical at x = 11/3.

Then, review the findings:
The line for equation 5 is a vertical line crossing the x-axis at approximately x = 3.67.

Finally, let me examine equation 6: 6) 2x + 4y = 30.

First, look closely:
- Equation 6 is 2x + 4y = 30.
- I need to convert this to slope-intercept form.

Next, find information:
2x + 4y = 30
4y = -2x + 30
y = (-1/2)x + 15/2
y = (-1/2)x + 7.5

This tells me the line has a slope of -1/2 and a y-intercept of 7.5.

Then, review the findings:
The line for equation 6 will cross the y-axis at (0,7.5) and have a slope of -1/2 (for every 2 units right, go 1 unit down).

To summarize the solution for each equation:

1) x + 4y = 0 → y = -1/4x (line through origin with slope -1/4)
2) x + y = -8 → y = -x - 8 (line with slope -1, y-intercept -8)
3) 5y = 3x → y = (3/5)x (line through origin with slope 3/5)
4) 6x = 3y + 3 → y = 2x - 1 (line with slope 2, y-intercept -1)
5) 3x = 18 - 7 → x = 11/3 (vertical line at x ≈ 3.67)
6) 2x + 4y = 30 → y = (-1/2)x + 7.5 (line with slope -1/2, y-intercept 7.5)

To graph each equation, you would plot the y-intercept (if applicable) and use the slope to find additional points, then draw a straight line through those points. For the vertical line in equation 5, you would simply draw a vertical line at x = 11/3.