Math Problem Statement
Find the equation of the line that models the relationship between x and y using the table of values: x = [15, 20, 25, 30, 35] and y = [100, 115, 130, 145, 160].
Solution
The table you have provided contains two columns of values, and , as follows:
x & y \\ \hline 15 & 100 \\ 20 & 115 \\ 25 & 130 \\ 30 & 145 \\ 35 & 160 \\ \end{array}$$ This data seems to represent a linear relationship between $$x$$ and $$y$$. To find the equation of the line that models this relationship, we can use the slope-intercept form of a line: $$y = mx + b$$ where $$m$$ is the slope and $$b$$ is the y-intercept. ### Step 1: Find the slope $$m$$ The slope of a line is given by the formula: $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ Using two points from the table, say $$(15, 100)$$ and $$(20, 115)$$, we can compute the slope: $$m = \frac{115 - 100}{20 - 15} = \frac{15}{5} = 3$$ ### Step 2: Find the y-intercept $$b$$ Now that we know the slope $$m = 3$$, we can use one of the points to solve for $$b$$. Using the point $$(15, 100)$$: $$100 = 3(15) + b$$ Solving for $$b$$: $$100 = 45 + b \implies b = 100 - 45 = 55$$ ### Step 3: Equation of the line Now that we have both the slope $$m = 3$$ and the intercept $$b = 55$$, the equation of the line is: $$y = 3x + 55$$ This equation models the relationship between $$x$$ and $$y$$ in the table. Would you like further details or clarifications? Here are five follow-up questions to explore this topic: 1. What would be the predicted $$y$$-value when $$x = 40$$ using the equation $$y = 3x + 55$$? 2. How can we determine if a relationship between two variables is truly linear? 3. How would the graph of the equation $$y = 3x + 55$$ look? 4. How can you calculate the slope if the relationship between $$x$$ and $$y$$ is not linear? 5. What happens to the line if we change the slope or intercept in the equation? **Tip:** When trying to fit data to a linear model, always verify the consistency of the slope between multiple pairs of points to ensure the relationship is linear.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Equations
Algebra
Formulas
Slope-intercept form: y = mx + b
Slope formula: m = (y2 - y1) / (x2 - x1)
Theorems
Linear Equation Theorem
Suitable Grade Level
Grades 8-10