Question

Write an equation of the line that passes through the point and has the given slope. Write the equation in slope-intercept form. $$(10,-10), m=\frac{2}{3}$$

Short answer

The equation of the line in slope-intercept form is \(y = \frac{2}{3}x - \frac{50}{3}\)

Step-by-step solution

Understanding the formula for slope-intercept form

The formula for any line in slope-intercept form is \(y = mx + b\) where m is the slope of the line and b is the y-intercept. This is the equation we need to derive from the given data.

Substitute the slope

We know the slope is \(m = \frac{2}{3}\). Substituting this into our equation, we now have \(y = \frac{2}{3}x + b\). Now we need to find the value of b.

Substitute point values to find y-intercept

We know the line passes through the point (10,-10), i.e., \(x=10\) and \(y=-10\). Substituting these values in for x and y in our equation, we can solve for b: \[ -10 = \frac{2}{3} * 10 + b \] \[ -10 = \frac{20}{3} + b \] \[ b = -10 - \frac{20}{3} = -\frac{50}{3} \]

Write final equation

We substitute the calculated value of b = -50/3 into the equation in slope-intercept form: \(y = \frac{2}{3}x - \frac{50}{3}\)