graph each system of equations and describe it as consistent and independent, consistent and dependent, or inconsistent

10. y+4x=12
3y=8-12x

11. -2x-3y=9
4x+6y=-18

12. 9x-2y=11
5x+4y=13

10 inconsistent

11 consistent dependent
12 consistent independent

To graph each system of equations and determine their consistency and independence, we can follow these steps:

10. y+4x=12, 3y=8-12x

First, let's convert the equations to slope-intercept form (y = mx + b) to make it easier to graph.
Equation 1: y + 4x = 12 can be rearranged to y = -4x + 12.
Equation 2: 3y = 8 - 12x can be simplified to y = (-4x + 8)/3.

Now, we will graph these two lines on the same coordinate plane:
Equation 1 (y = -4x + 12) has a y-intercept of 12 and a slope of -4. It can be plotted by marking the point (0, 12) on the y-axis and using the slope to find additional points.
Equation 2 (y = (-4x + 8)/3) has a y-intercept of 8/3 and a slope of -4/3. Plot the point (0, 8/3) and find more points using the slope.

After graphing the lines, we observe that they intersect at a single point (2, 4). This indicates that the system is consistent and independent. The solution to the system is x = 2 and y = 4.

11. -2x-3y=9, 4x+6y=-18

Let's convert the equations to slope-intercept form.

Equation 1: -2x - 3y = 9 can be rearranged to y = (-2x - 9)/3.
Equation 2: 4x + 6y = -18 can be simplified to y = (-2x - 3).

Now, we will graph these two lines on the same coordinate plane:
Equation 1 (y = (-2x - 9)/3) has a y-intercept of -3 and a slope of -2/3.
Equation 2 (y = (-2x - 3)) also has a y-intercept of -3 and a slope of -2.

After graphing the lines, we observe that they coincide, meaning they are the same line and overlap each other. This indicates that the system is consistent and dependent. The solution to the system is infinite solutions, as all points on the line satisfy both equations.

12. 9x - 2y = 11, 5x + 4y = 13

Let's convert the equations to slope-intercept form.

Equation 1: 9x - 2y = 11 can be rearranged to y = (9x - 11)/2.
Equation 2: 5x + 4y = 13 can be simplified to y = (-5x + 13)/4.

Now, we will graph these two lines on the same coordinate plane:
Equation 1 (y = (9x - 11)/2) has a y-intercept of -11/2 and a slope of 9/2.
Equation 2 (y = (-5x + 13)/4) has a y-intercept of 13/4 and a slope of -5/4.

After graphing the lines, we observe that they intersect at a single point, approximately (1.26, 2.63). This indicates that the system is consistent and independent. The solution to the system is x ≈ 1.26 and y ≈ 2.63.