Math Problem Statement

The equation of line g is 3x - 5y = -10. Line h, which is perpendicular to line g, includes the point (-5, 9). What is the equation of line h?

Solution

We are tasked with finding the equation of line hh, which is perpendicular to line gg and passes through the point (5,9)(-5, 9).

Step 1: Find the slope of line gg

The equation of line gg is: 3x5y=103x - 5y = -10 We need to rewrite this in slope-intercept form y=mx+by = mx + b to find the slope.

Rearranging the terms: 5y=3x10-5y = -3x - 10 y=35x+2y = \frac{3}{5}x + 2 So, the slope mgm_g of line gg is 35\frac{3}{5}.

Step 2: Find the slope of line hh

Since line hh is perpendicular to line gg, the slope of line hh, mhm_h, will be the negative reciprocal of mgm_g. The negative reciprocal of 35\frac{3}{5} is 53-\frac{5}{3}.

Step 3: Use the point-slope form to find the equation of line hh

Now that we have the slope mh=53m_h = -\frac{5}{3} and a point (5,9)(-5, 9) that lies on line hh, we use the point-slope form: yy1=m(xx1)y - y_1 = m(x - x_1) Substitute m=53m = -\frac{5}{3}, x1=5x_1 = -5, and y1=9y_1 = 9: y9=53(x+5)y - 9 = -\frac{5}{3}(x + 5) Simplifying: y9=53x253y - 9 = -\frac{5}{3}x - \frac{25}{3} y=53x253+9y = -\frac{5}{3}x - \frac{25}{3} + 9 Convert 9 to a fraction with a denominator of 3: y=53x253+273y = -\frac{5}{3}x - \frac{25}{3} + \frac{27}{3} y=53x+23y = -\frac{5}{3}x + \frac{2}{3}

Final Answer:

The equation of line hh in slope-intercept form is: y=53x+23y = -\frac{5}{3}x + \frac{2}{3}

Would you like any further details or explanations?

Here are 5 related questions to expand on this:

  1. How do you find the equation of a line parallel to a given line?
  2. What is the geometric significance of the slope being the negative reciprocal?
  3. How do you convert between different forms of a line equation (standard, point-slope, slope-intercept)?
  4. How can you determine if two lines are perpendicular using only their equations?
  5. Can you find the equation of a line parallel to line gg that passes through a different point?

Tip: When dealing with perpendicular lines, remember that their slopes are always negative reciprocals of each other.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Coordinate Geometry

Formulas

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

Theorems

Perpendicular lines have slopes that are negative reciprocals

Suitable Grade Level

High School