Math Problem Statement

Write an equation for a line passing through the point (6,5) that is parallel to y = (2/7)x + 3. Then write a second equation for a line passing through the given point that is perpendicular to the given line.

Solution

The problem asks for two equations of lines:

  1. Equation of a line parallel to the given line y=27x+3y = \frac{2}{7}x + 3 that passes through the point (6,5)(6,5).
  2. Equation of a line perpendicular to the same given line, passing through the same point.

Step 1: Find the equation of the parallel line.

The slope of the given line is m=27m = \frac{2}{7}. Since parallel lines have the same slope, the equation of the new line will also have a slope of 27\frac{2}{7}.

We will use the point-slope form of the equation of a line: yy1=m(xx1)y - y_1 = m(x - x_1) where m=27m = \frac{2}{7} and the point (x1,y1)=(6,5)(x_1, y_1) = (6,5).

Substituting: y5=27(x6)y - 5 = \frac{2}{7}(x - 6) Now simplify: y5=27x127y - 5 = \frac{2}{7}x - \frac{12}{7} y=27x127+5y = \frac{2}{7}x - \frac{12}{7} + 5 Convert 5 to a fraction with denominator 7: y=27x127+357y = \frac{2}{7}x - \frac{12}{7} + \frac{35}{7} y=27x+237y = \frac{2}{7}x + \frac{23}{7}

So, the equation of the parallel line is: y=27x+237y = \frac{2}{7}x + \frac{23}{7}

Step 2: Find the equation of the perpendicular line.

For a line perpendicular to y=27x+3y = \frac{2}{7}x + 3, the slope will be the negative reciprocal of 27\frac{2}{7}. The negative reciprocal of 27\frac{2}{7} is 72-\frac{7}{2}.

Using the same point-slope form with m=72m = -\frac{7}{2} and the point (6,5)(6, 5), we have: y5=72(x6)y - 5 = -\frac{7}{2}(x - 6) Simplify: y5=72x+21y - 5 = -\frac{7}{2}x + 21 y=72x+21+5y = -\frac{7}{2}x + 21 + 5 y=72x+26y = -\frac{7}{2}x + 26

So, the equation of the perpendicular line is: y=72x+26y = -\frac{7}{2}x + 26

Final Answer:

  • The equation of the line parallel to y=27x+3y = \frac{2}{7}x + 3 is y=27x+237y = \frac{2}{7}x + \frac{23}{7}.
  • The equation of the line perpendicular to y=27x+3y = \frac{2}{7}x + 3 is y=72x+26y = -\frac{7}{2}x + 26.

Would you like more details or have any questions?

5 Related Questions:

  1. How do you find the slope of a line from an equation in standard form?
  2. What is the relationship between the slopes of parallel and perpendicular lines?
  3. How do you convert from point-slope form to slope-intercept form?
  4. What is the general form of a linear equation?
  5. How do you graph a line given its equation in slope-intercept form?

Tip:

When working with fractions in equations, it helps to clear denominators early if possible to simplify calculations!

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Math Problem Analysis

Mathematical Concepts

Linear Equations
Slope-Intercept Form
Parallel and Perpendicular Lines

Formulas

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

Theorems

Parallel lines have the same slope.
Perpendicular lines have slopes that are negative reciprocals.

Suitable Grade Level

Grades 9-10