Finding the Linear Equation for the Data Set (x, y)

Math Problem Statement

Given the table with x and y values: (-12, 87), (-8, 59), (-4, 31), (0, 3), find the linear equation that fits the data.

From the table you've provided, there are pairs of $x$ and $y$ values, which suggest a possible linear relationship. The points are:

\hline x & y \\ \hline -12 & 87 \\ -8 & 59 \\ -4 & 31 \\ 0 & 3 \\ \hline \end{array}$$ ### Step 1: Find the slope $$m$$ The formula for the slope $$m$$ between two points $$(x_1, y_1)$$ and $$(x_2, y_2)$$ is: $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ Let's use the points $$(-12, 87)$$ and $$(0, 3)$$: $$m = \frac{3 - 87}{0 - (-12)} = \frac{-84}{12} = -7$$ ### Step 2: Find the equation of the line The equation of a line in slope-intercept form is: $$y = mx + b$$ We already know $$m = -7$$, so we can use one of the points to solve for $$b$$, the y-intercept. Let's use the point $$(0, 3)$$: $$3 = -7(0) + b \quad \Rightarrow \quad b = 3$$ Thus, the equation of the line is: $$y = -7x + 3$$ This equation represents the linear relationship between $$x$$ and $$y$$ based on the data you provided. Would you like more details or have any questions on this? Here are some related questions you might explore: 1. Can we verify the linear equation using the other points in the table? 2. How would you interpret the slope of $$-7$$ in a real-world context? 3. Can we predict the value of $$y$$ when $$x = 2$$? 4. What happens to $$y$$ as $$x$$ increases by 4 units based on this equation? 5. How can we graph this line using the given equation? **Tip:** The slope of a line tells you how steep the line is and whether it’s increasing or decreasing. A negative slope means the line decreases as you move from left to right.

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Algebra
Linear Equations

Slope formula: m = (y2 - y1) / (x2 - x1)
Equation of a line: y = mx + b

Slope-Intercept Form of a Linear Equation

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