Math Problem Statement

What's the answer

What's the answer

Solution

The table shows a representation of values for xx and yy. It seems like the task is to analyze the relationship between xx and yy. To proceed, I will:

  1. Identify if there’s a linear relationship between xx and yy.
  2. Determine the slope (mm) and y-intercept (bb) if it is linear.

Given Data:

  • For x=3x = -3, y=312=3.5y = 3 \frac{1}{2} = 3.5
  • For x=7x = 7, y=112=1.5y = -1 \frac{1}{2} = -1.5
  • For x=9x = 9, y=212=2.5y = -2 \frac{1}{2} = -2.5

Step 1: Check for Linear Relationship

Using the slope formula m=y2y1x2x1m = \frac{y_2 - y_1}{x_2 - x_1}:

  • Between (3,3.5)(-3, 3.5) and (7,1.5)(7, -1.5): m=1.53.57(3)=510=0.5m = \frac{-1.5 - 3.5}{7 - (-3)} = \frac{-5}{10} = -0.5

  • Between (7,1.5)(7, -1.5) and (9,2.5)(9, -2.5): m=2.5(1.5)97=12=0.5m = \frac{-2.5 - (-1.5)}{9 - 7} = \frac{-1}{2} = -0.5

The slope is consistent, so the relationship is linear.

Step 2: Find the Equation of the Line

Using the slope-intercept form y=mx+by = mx + b with m=0.5m = -0.5:

  • Substitute x=3x = -3, y=3.5y = 3.5 into y=mx+by = mx + b: 3.5=0.5(3)+b3.5 = -0.5(-3) + b 3.5=1.5+b    b=23.5 = 1.5 + b \implies b = 2

Thus, the equation of the line is: y=0.5x+2y = -0.5x + 2

Final Answer:

The relationship between xx and yy is described by the equation: y=0.5x+2y = -0.5x + 2

Let me know if you need further clarification!


5 Related Questions:

  1. How do you calculate the slope between two points?
  2. What are the steps to derive the slope-intercept form of a linear equation?
  3. Can you determine the y-intercept directly from a table of values?
  4. How do you verify if data points follow a linear relationship?
  5. What happens if the slope varies between pairs of points?

Tip: To check a linear relationship, always verify that the slope between consecutive points is consistent!

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Math Problem Analysis

Mathematical Concepts

Linear Relationships
Slope-Intercept Form
Algebra

Formulas

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

Theorems

Linear Equation Theorem

Suitable Grade Level

Grades 8-10