Question

In Exercises \(21-24,\) write a general linear equation for the line through the two points. $$(-2,1), \quad(2,-2)$$

Short answer

The general linear equation of the line through the points (-2,1) and (2,-2) is \(y = -3/4*x - 1/2.\)

Step-by-step solution

Calculate the Slope

Use the formula \(m = (y2-y1)/(x2-x1)\) to calculate the slope of the line. Here \(x1 = -2, y1 = 1, x2 = 2\) and \(y2 = -2.\) So, the slope \(m = ((-2) - 1) / (2 - (-2)) = -3/4.

Substitute in Slope-Intercept Form

Now that we have the slope, we can substitute this value into the slope-intercept form \(y = mx + b\). This gives us the equation \(y = -3/4*x + b.\)

Find the Y-Intercept

Substitute one of the given points into the equation to solve for \(b\). Here, let's use the point (2,-2). Substituting, we get -2 = -3/4*2 + b. Solving for \(b\) gives \(b = -1/2.\) So, \(b = -1/2.\)

Write the Final Equation

Substitute \(b = -1/2\) back into the equation \(y = mx+b\) to write the final equation of the line as \(y = -3/4*x - 1/2.\)