Question

Given the line 2x + 3y=6

(i) make the subject of the formula of 2x + 3y= 6
(ii) find the gradient of the line
(iii) find the coordinates of the point at which the line cuts the x-axis.​

Answer 1

(i) To make 2x + 3y = 6 the subject of the formula, we need to isolate y on one side of the equation. We can do this by subtracting 2x from both sides of the equation:

2x + 3y = 6

3y = 6 - 2x

y = (6 - 2x)/3

Therefore, the subject of the formula is y = (6 - 2x)/3.

(ii) The gradient of the line can be found by rearranging the equation into slope-intercept form, y = mx + b, where m is the gradient of the line.

2x + 3y = 6

3y = -2x + 6

y = (-2/3)x + 2

Comparing this to the slope-intercept form, we see that the gradient of the line is -2/3.

(iii) To find the coordinates of the point at which the line cuts the x-axis, we need to set y = 0 in the equation of the line and solve for x:

y = (-2/3)x + 2

0 = (-2/3)x + 2

2/3x = 2

x = 3

Therefore, the point at which the line cuts the x-axis has coordinates (3, 0).

Answer 2

(i) the formula with y as the subject is y = (6 - 2x) / 3.

(ii) the gradient of the line is -2/3.

(iii) the coordinates of the point at which the line cuts the x-axis are (3, 0).

Let's go through each of the requested steps step by step:

(i) Make the subject of the formula of 2x + 3y = 6:

To make y the subject of the formula, we want to isolate y on one side of the equation.

2x + 3y = 6

Subtract 2x from both sides:

3y = 6 - 2x

Now, divide both sides by 3 to solve for y:

y = (6 - 2x) / 3

So, the formula with y as the subject is y = (6 - 2x) / 3.

(ii) Find the gradient of the line:

The equation of the line is in the form Ax + By = C, where A is the coefficient of x, and B is the coefficient of y. The gradient (m) of the line can be found using the formula:

m = -A / B

In this case, A = 2 and B = 3. Plug these values into the formula:

m = -2 / 3

So, the gradient of the line is -2/3.

(iii) Find the coordinates of the point at which the line cuts the x-axis:

When a line cuts the x-axis, the y-coordinate of the point of intersection is 0. To find the x-coordinate, we can substitute y = 0 into the equation and solve for x:

2x + 3y = 6

2x + 3(0) = 6

2x = 6

Now, divide both sides by 2 to solve for x:

x = 6 / 2

x = 3

So, the coordinates of the point at which the line cuts the x-axis are (3, 0).