Math Problem Statement

Find the equation of a line perpendicular to -5x + y = -2 that passes through the point (5,4).

Solution

The question asks to find the equation of a line perpendicular to the given line 5x+y=2-5x + y = -2 and passing through the point (5,4)(5, 4).

Step-by-step solution:

  1. Rewrite the given equation:
    The line equation is 5x+y=2-5x + y = -2. Let's rewrite it in slope-intercept form y=mx+by = mx + b to identify its slope: y=5x2y = 5x - 2 So, the slope of this line is m1=5m_1 = 5.

  2. Find the slope of the perpendicular line:
    The slope of a line perpendicular to another is the negative reciprocal of the original line's slope.
    Therefore, the slope of the new line is: m2=15m_2 = -\frac{1}{5}

  3. Use the point-slope form:
    The point-slope form of a line is given by: yy1=m(xx1)y - y_1 = m(x - x_1) where (x1,y1)(x_1, y_1) is a point on the line, and mm is the slope. Using the point (5,4)(5, 4) and the slope m2=15m_2 = -\frac{1}{5}, we have: y4=15(x5)y - 4 = -\frac{1}{5}(x - 5)

  4. Simplify the equation:
    Distribute and simplify the equation: y4=15x+1y - 4 = -\frac{1}{5}x + 1 y=15x+5y = -\frac{1}{5}x + 5

So, the equation of the line perpendicular to 5x+y=2-5x + y = -2 and passing through (5,4)(5, 4) is: y=15x+5y = -\frac{1}{5}x + 5

Would you like more details or explanations? Here are a few related questions to consider:

  1. How do you convert any linear equation to slope-intercept form?
  2. Why is the slope of a perpendicular line the negative reciprocal?
  3. What are other ways to find the equation of a line besides point-slope form?
  4. How would you graph two perpendicular lines on the coordinate plane?
  5. What happens if the line is parallel instead of perpendicular?

Tip: To verify your work, always check that the slopes of perpendicular lines multiply to 1-1.

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Perpendicular Lines
Slope-Intercept Form

Formulas

Slope-Intercept Form: y = mx + b
Point-Slope Form: y - y1 = m(x - x1)
Negative Reciprocal of the Slope

Theorems

Slope of Perpendicular Lines: The product of the slopes of two perpendicular lines is -1.

Suitable Grade Level

Grades 9-10