line p has a gradient -2/3 and passes through point(-6,7).line Q passed through points (4,4) and (2,6).Find the coordinates of the point where the two lines intersect?

All_for_one

2 years ago

To find the intersection point of the two lines, let's first determine the equation of each line.

For Line \(p\) with a gradient of \(-\frac{2}{3}\) passing through \((-6, 7)\), its equation using the point-gradient form is:

\(y - y_1 = m(x - x_1)\), where \(m\) is the gradient and \((x_1, y_1)\) is the point on the line.

Substituting the values, we get:

\(y - 7 = -\frac{2}{3}(x + 6)\)
\(y = -\frac{2}{3}x + 4\)

For Line \(Q\) passing through \((4, 4)\) and \((2, 6)\), let's find its equation using the two-point form:

\(y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1)\)

Substituting the values:

\(y - 4 = \frac{6 - 4}{2 - 4}(x - 4)\)
\(y - 4 = -1(x - 4)\)
\(y = -x + 8\)

Now, to find the point of intersection, we'll solve these equations simultaneously:

\(-\frac{2}{3}x + 4 = -x + 8\)

Let's solve for \(x\):

\(-2x + 12 = -3x + 24\)
\(x = -12\)

Substitute \(x = -12\) into one of the equations (let's use \(y = -\frac{2}{3}x + 4\)):

\(y = -\frac{2}{3}(-12) + 4\)
\(y = 8\)

Therefore, the point of intersection for the two lines is \((-12, 8)\).