Write the equation of the line that passes through the points left bracket, minus, 4, comma, 5, right bracket(−4,5) and left bracket, 2, comma, minus, 4, right bracket(2,−4). Put your answer in fully simplified point-slope form, unless it is a vertical or horizontal line.
To find the equation of the line, we first need to find the slope of the line using the formula:
\(\text{slope} = \frac{{y_2 - y_1}}{{x_2 - x_1}}\)
where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the given points.
Using the given points \((-4,5)\) and \((2,-4)\), we have:
\(\text{slope} = \frac{{(-4) - 5}}{{2 - (-4)}} = \frac{{-9}}{{6}} = -\frac{3}{2}\)
Now we can use the point-slope form of the equation of a line:
\(y - y_1 = m(x - x_1)\)
where \(m\) is the slope, and \((x_1, y_1)\) is any point on the line. We can choose either of the given points to substitute into the equation.
Let's use the point \((-4,5)\):
\(y - 5 = -\frac{3}{2}(x - (-4))\)
Simplifying:
\(y - 5 = -\frac{3}{2}(x + 4)\)
Expanding:
\(y - 5 = -\frac{3}{2}x - 6\)
Adding 5 to both sides to isolate \(y\):
\(y = -\frac{3}{2}x - 1\)
Therefore, the equation of the line passing through the points \((-4,5)\) and \((2,-4)\) is \(y = -\frac{3}{2}x - 1\) in point-slope form.