Math Problem Statement

parallel to 5x+6y=-1 and passes through (3,2)

Solution

To find the equation of the line parallel to 5x+6y=15x + 6y = -1 and passing through the point (3,2)(3, 2), we need to follow these steps:

  1. Find the slope of the given line:
    The equation of the line is in standard form: 5x+6y=15x + 6y = -1.
    To find the slope, we first rewrite the equation in slope-intercept form, y=mx+by = mx + b, where mm is the slope.

    6y=5x16y = -5x - 1 y=56x16y = -\frac{5}{6}x - \frac{1}{6} The slope of this line is m=56m = -\frac{5}{6}.

  2. Use the point-slope form:
    Since parallel lines have the same slope, the slope of the new line is also m=56m = -\frac{5}{6}.
    Now, use the point-slope form of a line equation:

    yy1=m(xx1)y - y_1 = m(x - x_1) where (x1,y1)=(3,2)(x_1, y_1) = (3, 2) is the given point and m=56m = -\frac{5}{6}.

    Substituting the values: y2=56(x3)y - 2 = -\frac{5}{6}(x - 3)

  3. Simplify the equation: Distribute the slope on the right-hand side: y2=56x+156y - 2 = -\frac{5}{6}x + \frac{15}{6} y2=56x+52y - 2 = -\frac{5}{6}x + \frac{5}{2}

    Now, add 2 to both sides: y=56x+52+2y = -\frac{5}{6}x + \frac{5}{2} + 2 y=56x+92y = -\frac{5}{6}x + \frac{9}{2}

Thus, the equation of the line parallel to 5x+6y=15x + 6y = -1 and passing through (3,2)(3, 2) is:

y=56x+92y = -\frac{5}{6}x + \frac{9}{2}


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

Here are 5 related questions to explore this concept further:

  1. How do you convert from point-slope form to standard form?
  2. What is the difference between parallel and perpendicular lines in terms of slope?
  3. Can you find the equation of a line perpendicular to 5x+6y=15x + 6y = -1?
  4. What happens if a line passes through the origin?
  5. How do you find the distance between two parallel lines?

Tip: Parallel lines always have the same slope but different intercepts!

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Parallel Lines
Slope-Intercept Form
Point-Slope Form

Formulas

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

Theorems

Parallel lines have the same slope

Suitable Grade Level

Grades 9-11