Question

what is the equation of the line that passes through the point (-5, 5) and has a slope of \\(\frac{3}{5}\\)?

Explanation:

Step1: Recall point - slope form

The point - slope form of a line is given by \(y - y_1=m(x - x_1)\), where \((x_1,y_1)\) is a point on the line and \(m\) is the slope of the line.
Here, \(x_1=- 5\), \(y_1 = 5\) and \(m=\frac{3}{5}\).

Step2: Substitute values into point - slope form

Substitute \(x_1=-5\), \(y_1 = 5\) and \(m = \frac{3}{5}\) into the formula \(y - y_1=m(x - x_1)\):
\(y-5=\frac{3}{5}(x - (-5))\)
Simplify the right - hand side: \(y - 5=\frac{3}{5}(x + 5)\)

Step3: Convert to slope - intercept form (optional, but to get a more standard form)

Distribute \(\frac{3}{5}\) on the right - hand side:
\(y-5=\frac{3}{5}x+\frac{3}{5}\times5\)
\(y-5=\frac{3}{5}x + 3\)
Add 5 to both sides of the equation:
\(y=\frac{3}{5}x+3 + 5\)
\(y=\frac{3}{5}x+8\)

We can also leave it in point - slope form or convert to standard form \(Ax+By = C\). Let's convert to standard form:
Starting from \(y=\frac{3}{5}x + 8\), subtract \(\frac{3}{5}x\) from both sides:
\(-\frac{3}{5}x+y=8\)
Multiply through by 5 to clear the fraction:
\(-3x + 5y=40\) or \(3x-5y=-40\) (or we can keep it in slope - intercept form \(y=\frac{3}{5}x + 8\))

Answer:

The equation of the line in slope - intercept form is \(y=\frac{3}{5}x + 8\) (or in standard form \(3x-5y=-40\) or in point - slope form \(y - 5=\frac{3}{5}(x + 5)\))