Find the Linear Equation from a Table of Points in Slope-Intercept Form

Math Problem Statement

Find the equation of the linear function represented by the table below in slope-intercept form. x | y 1 | 4 2 | 3 3 | 2 4 | 1

Solution

We are asked to find the equation of the linear function represented by the table in slope-intercept form $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.

The given table of values is:

\hline x & y \\ \hline 1 & 4 \\ 2 & 3 \\ 3 & 2 \\ 4 & 1 \\ \hline \end{array}$$ ### Step 1: Find the slope $$m$$ The slope $$m$$ of a linear function is calculated using the formula: $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ Using the points $$(1, 4)$$ and $$(2, 3)$$ from the table: $$m = \frac{3 - 4}{2 - 1} = \frac{-1}{1} = -1$$ Thus, the slope $$m = -1$$. ### Step 2: Find the y-intercept $$b$$ To find $$b$$, substitute the slope $$m = -1$$ and one of the points (e.g., $$(1, 4)$$) into the slope-intercept form $$y = mx + b$$: $$4 = -1(1) + b$$ $$4 = -1 + b$$ $$b = 5$$ ### Step 3: Write the equation Now that we have $$m = -1$$ and $$b = 5$$, the equation of the linear function is: $$y = -x + 5$$ This is the required linear equation in slope-intercept form. Would you like further details or have any questions? Here are some related questions for practice: 1. How would the equation change if the y-intercept were different? 2. Can you explain how to find the slope from any two points in general? 3. What happens to the line if the slope $$m$$ were positive? 4. How can you verify if other points from the table satisfy the equation $$y = -x + 5$$? 5. How do you graph the equation $$y = -x + 5$$? **Tip:** Always pick two different points from a table to calculate the slope of a linear function, ensuring consistency in results.

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Math Problem Analysis

Mathematical Concepts

Linear Functions
Slope-Intercept Form
Algebra

Formulas

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

Theorems

Basic Linear Function Theorem

Suitable Grade Level

Grades 7-9

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