Math Problem Statement

Check the descriptions that apply for the given equations to determine if they are linear or non-linear.

Solution

This question asks you to classify equations as linear or non-linear. Let’s analyze each equation in detail to determine its category:

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

    • This is a linear equation because it can be written in the form y=mx+by = mx + b, where mm (the slope) is 12\frac{1}{2} and bb (the y-intercept) is 22.
  2. xy=3xy = 3:

    • This is a non-linear equation because the variables xx and yy are multiplied together, creating a product term. Linear equations only have terms where variables are added, subtracted, or scaled by constants.
  3. y=2xy = -\frac{2}{x}:

    • This is a non-linear equation because it contains a variable in the denominator (1x\frac{1}{x}), which introduces a reciprocal relationship.
  4. y=x2+3y = x^2 + 3:

    • This is a non-linear equation because the variable xx is squared (x2x^2), which creates a quadratic term.
  5. y=x+3y = x + 3:

    • This is a linear equation because it is in the form y=mx+by = mx + b, where m=1m = 1 and b=3b = 3.
  6. y=43xy = \frac{4}{3x}:

    • This is a non-linear equation because it contains a variable in the denominator (1x\frac{1}{x}).

Final Classification:

  • Linear:

    • y=12x+2y = \frac{1}{2}x + 2
    • y=x+3y = x + 3
  • Non-linear:

    • xy=3xy = 3
    • y=2xy = -\frac{2}{x}
    • y=x2+3y = x^2 + 3
    • y=43xy = \frac{4}{3x}

Let me know if you want further clarifications or explanations!


Related Questions:

  1. How do you rewrite equations in the form y=mx+by = mx + b to check if they are linear?
  2. What is the key difference between linear and quadratic equations?
  3. How do reciprocal terms (like 1x\frac{1}{x}) affect the graph of an equation?
  4. Can a product of variables (e.g., xy=3xy = 3) ever be considered linear? Why or why not?
  5. What are the real-world applications of classifying equations as linear or non-linear?

Tip:

To identify if an equation is linear, check if the highest power of the variable is 1 and if the variables are not multiplied together or appear in denominators.

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Math Problem Analysis

Mathematical Concepts

Linear Equations
Non-Linear Equations
Slope-Intercept Form
Reciprocal Relationships

Formulas

y = mx + b (slope-intercept form)

Theorems

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Suitable Grade Level

Grade 8