The line 3x + my = 3m cuts a triangular portion from the first quadrant whose area is 6. What is the value of m?

Well, it seems like our line is quite the artist, cutting out triangles in the first quadrant! But let me tell you, finding the value of m is no joke.

To find the area of a triangle, we need to know the length of the base and the height.

The base will be the x-coordinate where the line intersects the x-axis, so let's set y to 0 and solve for x:

3x + my = 3m
3x + m(0) = 3m
3x = 3m
x = m

So, the length of the base is m.

Now, let's find the height. The height will be the y-coordinate where the line intersects the y-axis, so let's set x to 0 and solve for y:

3(0) + my = 3m
my = 3m
y = 3

So, the height is 3.

Now, we can calculate the area of the triangle using the formula:

Area = (1/2) * base * height
6 = (1/2) * m * 3

Solving for m:
6 = (3/2) * m
m = 12/3
m = 4

Ta-dah! The value of m that cuts a triangular portion from the first quadrant with an area of 6 is 4. I hope that brings a smile to your face!

To find the value of m, we need to solve the problem step by step.

Step 1: Determine the points where the line intersects the axes.
To find the x-intercept of the line, set y equal to 0:
3x + my = 3m
3x + m(0) = 3m
3x = 3m
x = m

To find the y-intercept of the line, set x equal to 0:
3(0) + my = 3m
0 + my = 3m
my = 3m
y = 3

Therefore, the line intersects the x-axis at (m, 0) and the y-axis at (0, 3).

Step 2: Find the vertices of the triangular portion in the first quadrant.
The first quadrant is where both x and y are positive. Since the y-intercept is 3, the triangular portion is bounded by the x-axis, y-axis, and the line 3x + my = 3m.

To find the x-coordinate of the vertex on the x-axis, we set the y-coordinate to zero:
3x + my = 3m
3x + m(0) = 3m
3x = 3m
x = m

To find the y-coordinate of the vertex on the y-axis, we set the x-coordinate to zero:
3(0) + my = 3m
0 + my = 3m
my = 3m
y = 3

Therefore, the vertices of the triangular portion in the first quadrant are (0, 0), (m, 0), and (0, 3).

Step 3: Calculate the area of the triangular portion.
To find the area of a triangle, we can use the formula: A = 1/2 * base * height.

The base of the triangle is the length of the line segment connecting (0, 0) and (m, 0), which is m - 0 = m.

The height of the triangle is the length of the line segment connecting (0, 0) and (0, 3), which is 3 - 0 = 3.

Substituting these values into the formula, we have:
A = 1/2 * m * 3 = 3/2 * m

Given that the area of the triangle is 6, we can set up the equation:
3/2 * m = 6

Multiply both sides of the equation by 2/3:
m = 6 * 2/3
m = 12/3
m = 4

Therefore, the value of m is 4.

sketch a graph

(0,3) and (m , 0) are intercepts

area = (1/2) * 3 * m = 1.5 m
so
1.5 m = 6
m = 4

To find the value of m, we need to determine the equation of the line 3x + my = 3m and then calculate the area of the triangle formed in the first quadrant.

First, let's rearrange the equation of the line to solve for y:
my = -3x + 3m
y = (-3x + 3m)/m = -3x/m + 3

Now, let's find the x-intercept and y-intercept of the line by setting y = 0 and x = 0, respectively:
For the x-intercept:
0 = -3x/m + 3
3x/m = 3
x = m

For the y-intercept:
y = -3(0)/m + 3
y = 3

Now, we have the coordinates of two vertices of the triangle: (0, 3) and (m, 0). To find the third vertex, we need to find the point of intersection of the line with the x-axis.

Setting y = 0 in the line equation:
0 = -3x/m + 3
3x/m = 3
x = m

So, the third vertex is (m, 0).

Now, we can find the area of the triangle formed by these three vertices. We can use the formula for the area of a triangle:

Area = (1/2) * base * height

The base of the triangle is the distance between (0, 3) and (m, 0), which is m units. The height is the distance between (m, 0) and (0, 0), which is also m units.

Area = (1/2) * m * m = m^2/2

Given that the area of the triangle is 6, we can set up the equation:

m^2/2 = 6

Now, we can solve for m:

m^2 = 12
m = ±√12
m = ±2√3

Therefore, the value of m can be either 2√3 or -2√3.