Math Problem Statement

Write the linear equation that gives the rule for this table: x | y ---|--- 4 | 9 5 | 10 6 | 11 7 | 12 Write your answer as an equation with y first, followed by an equals sign.

Solution

To find the linear equation that represents the relationship between xx and yy in the given table, we can follow these steps:

Table Analysis

The table provided is:

x & y \\ \hline 4 & 9 \\ 5 & 10 \\ 6 & 11 \\ 7 & 12 \\ \end{array}$$ ### Step 1: Find the slope (m) The formula for the slope $$m$$ of a linear equation $$y = mx + b$$ is given by: $$m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$$ Using the first two points $$(4, 9)$$ and $$(5, 10)$$: $$m = \frac{10 - 9}{5 - 4} = \frac{1}{1} = 1$$ ### Step 2: Find the y-intercept (b) Now that we have the slope, we can use one of the points to find the y-intercept $$b$$. Using the point $$(4, 9)$$: $$y = mx + b$$ Substitute $$m = 1$$, $$x = 4$$, and $$y = 9$$: $$9 = 1(4) + b$$ $$b = 9 - 4 = 5$$ ### Step 3: Write the equation Now that we have $$m = 1$$ and $$b = 5$$, the equation of the line is: $$y = x + 5$$ This is the linear equation that represents the data in the table. Would you like more details, or do you have any questions about this solution? Here are 5 related questions you might explore: 1. What happens to the equation if the slope was negative instead? 2. How would the equation change if the y-intercept were zero? 3. How do you determine if a set of points forms a linear relationship? 4. What are the implications of having a slope of zero? 5. Can a linear equation ever have a variable exponent greater than 1? **Tip:** To verify your linear equation, you can substitute each $$x$$ value from the table into the equation and check if you get the corresponding $$y$$ value.

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Slope-Intercept Form

Formulas

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

Theorems

Slope-Intercept Form

Suitable Grade Level

Grades 6-8

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