Question

3 The line through (2, -4) and (x, 1) has a slope of -5. Determine the equation of the line and the value of x.

Answer 1

To determine the equation of the line through the points (2, -4) and (x, 1) with a slope of -5, you can use the point-slope form of the equation of a line:

y - y_1 = m(x - x_1)

where (x1, y1) is a point on the line, m is the slope, and (x, y) are the coordinates of any other point on the line.

In this case, you have the point (2, -4) and the slope m = -5:

y - (-4) = -5(x - 2)

Now, simplify the equation:

y + 4 = -5(x - 2)

Next, distribute the -5 on the right side:

y + 4 = -5x + 10

Now, isolate y by subtracting 4 from both sides:

y = -5x + 6

So, the equation of the line is y = -5x + 6.

To find the value of x, you can use the fact that the line passes through the point (x, 1). Substitute this point into the equation:

1 = -5x + 6

Now, solve for x:

5x = 6 - 1

5x = 5

x = 1

So, the value of x is 1.

Answer 2

To find the equation of the line, we need to use the point-slope form of a linear equation, which is:

y - y1 = m(x - x1),

where (x1, y1) is a point on the line, and m is the slope of the line.

Given that the line passes through (2, -4) and (x, 1), and has a slope of -5, we can substitute these values into the point-slope form.

Let's substitute (2, -4) into the equation:

y - (-4) = -5(x - 2).

Simplifying, we have:

y + 4 = -5x + 10.

Next, let's substitute (x, 1) into the equation:

1 + 4 = -5x + 10.

Simplifying, we have:

5 = -5x + 10.

Now, let's solve for x:

-5x = 5.

Dividing both sides by -5, we get:

x = -1.

So, the value of x is -1.

Finally, let's rearrange the equation in slope-intercept form (y = mx + b), where b is the y-intercept:

y + 4 = -5x + 10.

Subtract 4 from both sides:

y = -5x + 6.

Therefore, the equation of the line passing through (2, -4) and (x, 1) with a slope of -5 is y = -5x + 6, and the value of x is -1.