Math Problem Statement

The lines shown below are perpendicular. Find the slope-intercept form of the equation of line y2.

Solution

The image presents two perpendicular lines, and the task is to find the slope-intercept form of the equation of line y2y_2, based on the equation given for line y1y_1. The equation of y1y_1 is:

y1=3x4y_1 = 3x - 4

Key Information:

  • The lines are perpendicular, meaning the product of their slopes must equal 1-1.
  • The slope of y1y_1 is 33 (since the equation is in the form y=mx+by = mx + b, where mm is the slope).
  • For perpendicular lines, if the slope of one line is m1m_1, the slope of the other line is m2=1m1m_2 = -\frac{1}{m_1}.

Thus, for line y2y_2, the slope m2m_2 is:

m2=13m_2 = -\frac{1}{3}

The equation of line y2y_2 is also in the form y=mx+by = mx + b. Now, to find the y-intercept b2b_2, we can use the point where the two lines intersect, which is given as (3,5)(3, 5). Using this point in the equation y2=13x+b2y_2 = -\frac{1}{3}x + b_2:

5=13(3)+b25 = -\frac{1}{3}(3) + b_2 5=1+b25 = -1 + b_2 b2=6b_2 = 6

Thus, the equation of y2y_2 is:

y2=13x+6y_2 = -\frac{1}{3}x + 6

Let me know if you would like further clarification or additional details.

Related Questions:

  1. How do you find the equation of a line if the slope and a point on the line are given?
  2. What is the significance of perpendicular lines in terms of their slopes?
  3. How can you verify the intersection point of two lines algebraically?
  4. What method would you use to find the distance between two lines?
  5. How do parallel and perpendicular slopes differ in geometry?

Tip:

Always remember, for two lines to be perpendicular, their slopes must multiply to 1-1. This relationship helps in quickly identifying the slope of the perpendicular line.

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Perpendicular Lines

Formulas

Slope-intercept form: y = mx + b
Perpendicular slope: m1 * m2 = -1

Theorems

Slopes of perpendicular lines theorem

Suitable Grade Level

Grades 8-10