Math Problem Statement

Line p has an equation of y = 5/3x - 4. Line q includes the point (-10, -3) and is perpendicular to line p. What is the equation of line q?

Solution

To solve this problem, let's go through the steps required to find the equation of line qq, which is perpendicular to line pp and passes through the point (10,3)(-10, -3).

Given Information

  • Equation of line pp: y=53x4y = \frac{5}{3}x - 4
  • Line qq passes through the point (10,3)(-10, -3) and is perpendicular to line pp.

Steps to Solution

  1. Identify the Slope of Line pp:

    • Line pp is in slope-intercept form y=mx+by = mx + b, where mm is the slope.
    • For line pp, the slope mp=53m_p = \frac{5}{3}.
  2. Determine the Slope of Line qq:

    • Since line qq is perpendicular to line pp, its slope mqm_q will be the negative reciprocal of mpm_p.
    • Therefore, mq=35m_q = -\frac{3}{5}.
  3. Use the Point-Slope Form to Find the Equation of Line qq:

    • We have the point (10,3)(-10, -3) and the slope 35-\frac{3}{5}.
    • The point-slope form of a line is: yy1=m(xx1)y - y_1 = m(x - x_1) Substituting x1=10x_1 = -10, y1=3y_1 = -3, and m=35m = -\frac{3}{5}: y(3)=35(x(10))y - (-3) = -\frac{3}{5}(x - (-10)) Simplify the equation: y+3=35(x+10)y + 3 = -\frac{3}{5}(x + 10)
  4. Simplify to Slope-Intercept Form y=mx+by = mx + b:

    • Distribute the slope on the right side: y+3=35x3510y + 3 = -\frac{3}{5}x - \frac{3}{5} \cdot 10
    • Calculate 3510=6-\frac{3}{5} \cdot 10 = -6: y+3=35x6y + 3 = -\frac{3}{5}x - 6
    • Subtract 3 from both sides to isolate yy: y=35x9y = -\frac{3}{5}x - 9

Final Answer

The equation of line qq in slope-intercept form is: y=35x9y = -\frac{3}{5}x - 9

Would you like further details on any step, or do you have any questions?

Here are some related questions to expand your understanding:

  1. How do you find the equation of a line parallel to a given line?
  2. What is the significance of the negative reciprocal in perpendicular slopes?
  3. How would you rewrite the equation if the point given was different?
  4. Can you convert the equation into standard form?
  5. What happens to the line if you change the slope while keeping the same point?

Tip: When working with perpendicular lines, remember that their slopes are always negative reciprocals. This concept is useful in geometry and analytic geometry problems.

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Math Problem Analysis

Mathematical Concepts

Algebra
Coordinate Geometry
Slope-Intercept Form
Perpendicular Lines

Formulas

Slope-intercept form: y = mx + b
Point-slope form: y - y1 = m(x - x1)

Theorems

Perpendicular slopes theorem (negative reciprocal relationship)

Suitable Grade Level

Grade 9