The equation of a line is y= 2/3x + 5. What is an equation of the line that is perpendicular to the given line and that passes through the point (4,2)?

Answer 1

y = (2/3)x + 5 is the linear equation for the current line.

We want to determine the equation of a line that is PERPENDICULAR to that line, and passes through the point (4,2).

First:

We can determine the slope quite simply by taking the original line's slope & finding the OPPOSITE RECIPROCAL of that. *The opposite' means that you will switch the sign. If it was positive, then the new slope wiil be negative. If it was negative, then the new slope wiil be positive. 'Reciprocal' means to take the original value and FLIP THE FRACTION!!

For a quick example, if our original equation showed a slope of 4/5, the "OPPOSITE RECIPROCAL" of that would be -5/4. We changed the sign from + to - and flipped the fraction over. IF 2 LINES ARE PERPENDICULAR, THEIR SLOPES WILL ALWAYS BE OPPOSITE RECIPROCALS OF EACH OTHER.

So, for your specific question, the slope of the original line is 2/3 (or +2/3). We need to find the opposite reciprocal of +2/3. Change it from positive to negative and then flip the fraction. The slope for the new line will be -3/2.

Now, it also must pass through the point (4,2). Plot that point on a graph and then find several points to the left of it and to the right of it, by counting the slope. Eventually, we wil run into the y-intercept and then we can create the equation.

The point on our new line directly to the right of (4,2) will be [(4 + 2), (2 - 3)] = (6,-1). One more point to the right of that -- (6 + 2, -1-3) = (8,-4). Now let's find some points to the left of (4,2). We do this by counting the slope in reverse. (4 - 2, 2 + 3) = (2,5). Do it again.... (2 - 2, 5 + 3) = (0,8). That is our y-intercept (or our 'b' value).

Equation of the perpendicular line has a slope of -3/2 (m = -3/2) and y-intercept of 8 (b = 8).

y = mx + b so, y = -3/2x + 8

Answer 2

Using point-slope form from Algebra 1

(y - 2) = (-3/2)(x - 4)