Math Problem Statement

What is the equation of this line based on the graph provided?

Solution

The problem shown in the image asks for the equation of the line represented on the graph. Let's go through the steps to find the equation of this line.

Step 1: Identify Two Points on the Line

From the graph, the line passes through two clear points:

  • Point 1: (0,2)(0, 2)
  • Point 2: (3,0)(3, 0)

Step 2: Calculate the Slope (m)

The slope mm of a line is given by the formula: m=y2y1x2x1m = \frac{y_2 - y_1}{x_2 - x_1} Using the points (0,2)(0, 2) and (3,0)(3, 0), we can substitute: m=0230=23m = \frac{0 - 2}{3 - 0} = \frac{-2}{3} Thus, the slope m=23m = -\frac{2}{3}.

Step 3: Use the Slope-Intercept Form

The equation of a line in slope-intercept form is: y=mx+by = mx + b We already know m=23m = -\frac{2}{3}, and we can use the point (0,2)(0, 2), where the line crosses the y-axis, to find bb, the y-intercept.

Since the y-intercept is the point where x=0x = 0, we can see that b=2b = 2.

Final Equation

Substituting m=23m = -\frac{2}{3} and b=2b = 2 into the slope-intercept form: y=23x+2y = -\frac{2}{3}x + 2

So, the equation of the line is: y=23x+2\boxed{y = -\frac{2}{3}x + 2}

Let me know if you need any clarifications or further details.

Questions to Explore:

  1. What is the slope of a horizontal line?
  2. How does changing the slope affect the steepness of a line?
  3. What happens to the equation of a line if the slope is zero?
  4. How can you find the slope of a line given only its graph?
  5. Can a line have a positive y-intercept and a negative slope?

Tip: Always use two clear points to calculate the slope to avoid mistakes!

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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

Linear Equation Theorem

Suitable Grade Level

Grades 6-8