Math Problem Statement

Find the equation of the line given the graph with two points (-4, 6) and (2, -3).

Solution

To find the equation of the line, we need to determine both the slope (denoted as ?\text{?}) and the y-intercept (the constant term in the equation y=mx+cy = mx + c).

Step 1: Identify the slope

The slope of the line can be calculated by choosing two points on the line. From the image, the points (4,6)(-4, 6) and (2,3)(2, -3) are on the line.

The slope mm is given by the formula: m=y2y1x2x1m = \frac{y_2 - y_1}{x_2 - x_1} Substituting the coordinates of the points: m=362(4)=96=32m = \frac{-3 - 6}{2 - (-4)} = \frac{-9}{6} = -\frac{3}{2}

So, the slope is 32-\frac{3}{2}.

Step 2: Identify the y-intercept

The y-intercept is where the line crosses the y-axis. From the graph, it appears the line crosses the y-axis at y=0y = 0.

Step 3: Write the equation

Substituting the slope and y-intercept into the equation y=mx+cy = mx + c: y=32x+0y = -\frac{3}{2}x + 0 This simplifies to: y=32xy = -\frac{3}{2}x

This is the equation of the line.

Would you like more details on any part of this process, or do you have any further questions?

Here are some follow-up questions you might find interesting:

  1. How do you calculate the slope when given two arbitrary points?
  2. What is the significance of the y-intercept in the equation of a line?
  3. Can a line have a slope of 0 or undefined? What do these cases represent?
  4. How can you determine the equation of a line if it is parallel or perpendicular to another line?
  5. What changes in the equation when the line is shifted up or down on the graph?

Tip: Always check the points you use for calculating the slope to ensure they are accurate; even a small mistake in the coordinates can lead to incorrect results.

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Slope-Intercept Form

Formulas

Slope formula: m = (y2 - y1) / (x2 - x1)
Equation of a line: y = mx + c

Theorems

Linear Equation

Suitable Grade Level

Grades 8-10