Math Problem Statement

Determine the equation of a line based on its graph.

Solution

To determine the equation of the line shown in the graph, let's proceed with the following steps:

  1. Identify Two Points on the Line:
    It looks like the line passes through two specific points on the grid, which I will estimate based on the image.

    • One point seems to be at (0,2)(0, 2), where the line intersects the y-axis.
    • Another point appears to be at (3,1)(3, -1).
  2. Calculate the Slope:
    The slope mm of a line passing through two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is calculated by: m=y2y1x2x1m = \frac{y_2 - y_1}{x_2 - x_1} Using the points (0,2)(0, 2) and (3,1)(3, -1): m=1230=33=1m = \frac{-1 - 2}{3 - 0} = \frac{-3}{3} = -1

  3. Formulate the Equation:
    Since the y-intercept (the point where the line crosses the y-axis) is 22, the equation of the line is in the slope-intercept form y=mx+by = mx + b, where m=1m = -1 and b=2b = 2.

    Thus, the equation of the line is: y=x+2y = -x + 2

Would you like further details or have any questions about this solution?


Here are some related questions to consider:

  1. How would the equation change if the slope were positive?
  2. What is the importance of the y-intercept in the line's equation?
  3. How can you determine if two lines are parallel by comparing their equations?
  4. How would you write the equation of a line if only one point and the slope are given?
  5. How does changing the slope affect the angle of the line relative to the x-axis?

Tip: When finding the slope, always remember to carefully check the points’ coordinates to ensure accuracy in calculations.

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Slope-Intercept Form

Formulas

Slope formula: m = (y2 - y1) / (x2 - x1)
Slope-intercept form: y = mx + b

Theorems

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Suitable Grade Level

Grades 7-9