What is the slope of a line that is perpendicular to the line given by the equation 2y + 3x = 1?

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To find the slope of a line that is perpendicular to the line represented by the equation 2y + 3x = 1, we first need to rewrite this equation in the slope-intercept form, which is y = mx + b, where m stands for the slope.

Starting with the equation:

2y + 3x = 1

We will isolate y by performing the following steps:

  1. Subtract 3x from both sides:
2y = -3x + 1
  1. Now, divide every term by 2 to solve for y:
y = -rac{3}{2}x + rac{1}{2}

From this equation, we can see that the slope (m) of the line is -3/2. In the case of perpendicular lines, the slopes have a special relationship: if two lines are perpendicular, the product of their slopes equals -1.

Let’s denote the slope of the line perpendicular to our original line as mperpendicular. The relationship can be expressed mathematically as:

m 	imes mperpendicular = -1

Substituting for m:

-rac{3}{2} 	imes mperpendicular = -1

To isolate mperpendicular, we can multiply both sides by -1:

rac{3}{2} 	imes mperpendicular = 1

Next, we’ll divide both sides by 3/2 or multiply by its reciprocal, which is 2/3:

mperpendicular = 1 	imes rac{2}{3} = rac{2}{3}

Therefore, the slope of the line that is perpendicular to the given line 2y + 3x = 1 is rac{2}{3}.

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