I'm confused about graphing or solving the linear equations for slope/intercept.

Here is what I solved:
3x+2y=-6 X-4y=-16

2y= -3x-6 -4y=1x-16

Y=-3/2+3 Y=+1/4+4
-------------
My answer for graphing the systems for the lines to intersect...well, they didn't intersect. But the answer is (4,-1)

Can someone explain my problem? And help explain how to solve it?

try playing around here, where I have plotted the first pair...

http://www.wolframalpha.com/input/?i=plot+3x%2B2y%3D-6,+x-4y%3D-16

since your lines have different slopes, the have to intersect. You must have plotted them wrong.

3x+2y = -6
x-4y = -16

2y = -3x-6
-4y = -x-16

You went wrong here. Why did you just drop the x stuff?
Y= -3/2 x - 3
y = 1/4 x + 4

Now equate the two expressions for y:

-3/2 x - 3 = 1/4 x + 4
-7/4 x = 7
x = -4
so, y=3

If the answer is (4,-1) you have some typos in the original equations.

It seems like there may be a slight error in the equations you provided. Let's go through the process step by step to help you solve and graph the system of linear equations correctly.

First, let's go back to the original equations:
3x + 2y = -6 (Equation 1)
x - 4y = -16 (Equation 2)

To solve the system of linear equations, we can use the method of substitution or elimination. Let's use the substitution method in this case.

Step 1: Solve one equation for one variable
Let's solve Equation 2 for x:
x = 4y - 16 (Equation 3)

Step 2: Substitute the expression in Equation 3 into Equation 1
Substituting Equation 3 into Equation 1, we get:
3(4y - 16) + 2y = -6

Simplifying the equation:
12y - 48 + 2y = -6
14y - 48 = -6
14y = 42
y = 42 / 14
y = 3

Step 3: Substitute the value of y back into one of the original equations to find x
Using Equation 2, we substitute y = 3:
x - 4(3) = -16
x - 12 = -16
x = -16 + 12
x = -4

So the solution of the system of equations is x = -4 and y = 3. Therefore, the point of intersection is (-4, 3).

Now let's move on to graphing the equations to visualize the solution:

To graph Equation 1 (3x + 2y = -6), rewrite it in slope-intercept form (y = mx + b):
2y = -3x - 6
y = (-3/2)x - 3

To plot the graph, start with the y-intercept, which is -3, and then use the slope -3/2 to find another point. The slope -3/2 means for every 2 units you move to the right on the x-axis (increase in x), you move down 3 units on the y-axis (decrease in y). Connect the two points and extend the line:

|
3 | .
| .
0 | .
| .
-3 |__________________
0 -3 -6 -9 -12
x-axis

For Equation 2 (x - 4y = -16), rewrite it in slope-intercept form:
x = 4y - 16

To graph this equation, start by setting y to zero and solve for x:
x = 4(0) - 16
x = -16

So the x-intercept is -16. Then, find another point using the slope of 4/1, which means for every 1 unit increase in y, there is a 4 unit increase in x. Connect the two points and extend the line:

|
4 | .
| .
0 | .
| .
-16 |__________________
0 4 8 12 16
x-axis

From the graph, you can see that the lines intersect at the point (-4, 3), which confirms the solution we found algebraically.

I hope this explanation helps you understand how to solve and graph a system of linear equations! Let me know if you have any further questions.

To solve the system of linear equations and find the point of intersection, let's go step by step:

Step 1: Start with the given equations:
3x + 2y = -6 (Equation 1)
x - 4y = -16 (Equation 2)

Step 2: Solve each equation for y to get the slope-intercept form (y = mx + b) where m is the slope and b is the y-intercept:
Equation 1:
3x + 2y = -6
2y = -3x - 6
y = (-3/2)x - 3 (Equation 3)

Equation 2:
x - 4y = -16
-4y = -x - 16
y = (1/4)x + 4 (Equation 4)

Step 3: Now we have two equations in slope-intercept form. We can compare the slopes of the two lines to determine if they intersect or not:
The slope of Equation 3 (m1) is -3/2.
The slope of Equation 4 (m2) is 1/4.

Since the slopes are different (-3/2 ≠ 1/4), the lines are not parallel, and therefore, they intersect.

Step 4: To find the point of intersection, we can set Equation 3 equal to Equation 4 and solve for x:
(-3/2)x - 3 = (1/4)x + 4

Step 5: Solve for x:
(-3/2 - 1/4)x = 4 + 3
(-6/4 - 1/4)x = 7
(-7/4)x = 7
x = -4

Step 6: Substitute the value of x back into either equation (Equation 1 or Equation 2) to solve for y:
Using Equation 2:
x - 4y = -16
-4 - 4y = -16
-4y = -16 + 4
-4y = -12
y = -12/(-4)
y = 3

Therefore, the point of intersection is (-4, 3).

It appears there was an error in your previous calculations. The correct answer for the point of intersection is (-4, 3), not (4, -1).