Math Problem Statement

Which equation represents the line shown in the accompanying diagram?

Solution

To determine which equation represents the line shown in the diagram, let's analyze the graph step-by-step.

Step 1: Determine the Slope

The slope (mm) of a line can be determined by finding the change in yy over the change in xx (Δy/Δx\Delta y / \Delta x).

The line intersects the graph at points (3,0)(-3, 0) and (0,3)(0, 3):

  • From (3,0)(-3, 0) to (0,3)(0, 3):
    • The rise (Δy\Delta y) is 33.
    • The run (Δx\Delta x) is 33.

Thus, the slope of the line is: m=ΔyΔx=33=1m = \frac{\Delta y}{\Delta x} = \frac{3}{3} = 1

Step 2: Identify the y-Intercept

The yy-intercept is the point where the line crosses the yy-axis. The line crosses at y=3y = 3.

Step 3: Write the Equation of the Line

The equation of a line in slope-intercept form is: y=mx+by = mx + b Where:

  • mm is the slope.
  • bb is the yy-intercept.

From the graph:

  • The slope m=1m = 1.
  • The y-intercept b=3b = 3.

Thus, the equation of the line is: y=x+3y = x + 3

Step 4: Match with the Given Choices

The choices given are:

  1. y=2x+3y = 2x + 3
  2. y=3x+12y = 3x + \frac{1}{2}
  3. y=12x+3y = \frac{1}{2}x + 3
  4. y=3x+2y = 3x + 2

None of these match y=x+3y = x + 3, which is unexpected, indicating I may have missed something.

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Graphing

Formulas

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

Theorems

Slope-Intercept Theorem

Suitable Grade Level

Grades 8-10