Math Problem Statement
Is the relationship between x and y linear or non-linear based on the table provided?
Solution
The table in the image shows pairs of and values:
\hline x & y \\ \hline -4 & -6 \\ -3 & -3 \\ -2 & 0 \\ -1 & 3 \\ \hline \end{array}$$ ### To determine if the relationship is linear: A relationship is **linear** if the rate of change of $$y$$ with respect to $$x$$ (the slope) is constant. 1. Compute the change in $$y$$ for each unit change in $$x$$: - From $$x = -4$$ to $$x = -3$$: $$y$$ changes from $$-6$$ to $$-3$$ → $$\Delta y = -3 - (-6) = 3$$. - From $$x = -3$$ to $$x = -2$$: $$y$$ changes from $$-3$$ to $$0$$ → $$\Delta y = 0 - (-3) = 3$$. - From $$x = -2$$ to $$x = -1$$: $$y$$ changes from $$0$$ to $$3$$ → $$\Delta y = 3 - 0 = 3$$. Since the change in $$y$$ is consistent (always 3) for each unit change in $$x$$, the relationship is **linear**. ### Now for the slope (rate of change): The slope $$m$$ is calculated as the ratio of the change in $$y$$ to the change in $$x$$: $$m = \frac{\Delta y}{\Delta x} = \frac{3}{1} = 3$$ Therefore, the equation of the line can be written as $$y = 3x + b$$, where $$b$$ is the y-intercept. Using one of the points, say $$(-4, -6)$$, to find $$b$$: $$-6 = 3(-4) + b \implies -6 = -12 + b \implies b = 6$$ So, the equation is $$y = 3x + 6$$. Let me know if you want any further details! Here are some related questions you might find helpful: 1. How can you verify if other relationships are linear or non-linear? 2. How do you graph a linear equation like $$y = 3x + 6$$? 3. How do you find the slope if the changes in $$x$$ are not constant? 4. What happens if the slope between points changes? 5. How can you convert a non-linear equation to a linear form? **Tip:** Always check for a constant rate of change to determine if a relation is linear!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Relationships
Rate of Change
Slope
Formulas
Slope formula: m = Δy / Δx
Linear equation: y = mx + b
Theorems
Slope Theorem
Suitable Grade Level
Grades 7-10