Math Problem Statement

The following graph contains line l and point M. Line l contains points (5, 1) and (0, 3). Point M is located at (-4, -3). Find another point with integral coordinates that lies on a line parallel to line l and passes through point M.

Solution

Let's break down this problem:

Given:

  • Line ll passes through the points (5,1)(5, 1) and (0,3)(0, 3).
  • The point MM is at (4,3)(-4, -3), and we need to find another point on a line parallel to ll that passes through MM, and has integer coordinates.

Step 1: Find the slope of line ll

The slope of a line passing through two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is given by: m=y2y1x2x1m = \frac{y_2 - y_1}{x_2 - x_1} For the points (5,1)(5, 1) and (0,3)(0, 3), the slope of line ll is: m=1350=25m = \frac{1 - 3}{5 - 0} = \frac{-2}{5}

Step 2: Find the equation of the line parallel to ll and passing through M(4,3)M(-4, -3)

A line parallel to ll will have the same slope, m=25m = -\frac{2}{5}. The equation of a line with slope mm passing through point (x1,y1)(x_1, y_1) is: yy1=m(xx1)y - y_1 = m(x - x_1) Substituting m=25m = -\frac{2}{5}, x1=4x_1 = -4, and y1=3y_1 = -3, we get: y+3=25(x+4)y + 3 = -\frac{2}{5}(x + 4) Simplifying: y+3=25x85y + 3 = -\frac{2}{5}x - \frac{8}{5} y=25x853y = -\frac{2}{5}x - \frac{8}{5} - 3 y=25x235y = -\frac{2}{5}x - \frac{23}{5}

Step 3: Find a point with integer coordinates

To find integer coordinates, we substitute the given answer choices into the equation of the parallel line:

  1. For (2,2)(-2, 2): y=25(2)235=45235=195y = -\frac{2}{5}(-2) - \frac{23}{5} = \frac{4}{5} - \frac{23}{5} = -\frac{19}{5} Not an integer.

  2. For (9,1)(-9, -1): y=25(9)235=185235=55=1y = -\frac{2}{5}(-9) - \frac{23}{5} = \frac{18}{5} - \frac{23}{5} = -\frac{5}{5} = -1 This works since y=1y = -1.

Thus, the point (9,1)(-9, -1) lies on the line parallel to ll and passes through MM.

Final Answer:

The correct point is (9,1)(-9, -1).

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

Here are 5 related questions to consider:

  1. How would you find the equation of the line ll in slope-intercept form?
  2. What is the significance of finding the slope of a line for parallel lines?
  3. How can we determine if two lines are perpendicular based on their slopes?
  4. What is the general form of the equation of a line, and how is it derived from the slope-intercept form?
  5. How could you check if a point lies on a given line using its equation?

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

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Math Problem Analysis

Mathematical Concepts

Coordinate Geometry
Algebra
Parallel Lines

Formulas

Slope Formula: m = (y2 - y1) / (x2 - x1)
Point-Slope Form: y - y1 = m(x - x1)

Theorems

Concept of Parallel Lines

Suitable Grade Level

Grades 8-10