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To understand how to write an equation of a line, we start by calculating its slope. The slope of a line indicates its steepness and direction. We use the slope formula to determine this value: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]In this formula, - \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of two distinct points on the line. - \(m\) represents the slope.For our exercise, the points given are \((5, 12)\) and \((6, -2)\). Plugging these points into the formula gives:\[ m = \frac{-2 - 12}{6 - 5} = \frac{-14}{1} = -14 \]This result tells us that the line moves downward with a steepness of 14 units downward for every unit it moves to the right. It's a negative slope, which slope means the line decreases as it moves from left to right.

Once we've determined the slope, we can use the **Point-Slope Form** of a line equation, useful for writing the equation of a line with a known slope and a point on the line. The formula is:\[ y - y_1 = m(x - x_1) \] Where:- \(m\) is the slope, - \((x_1, y_1)\) is a point on the line.Using the slope calculated as -14 and selecting point \((5, 12)\), we substitute into the formula:\[ y - 12 = -14(x - 5) \]This equation represents the line in its point-slope form. It shows how the y-coordinate changes as the x-coordinate changes, based on the slope value. This form is particularly useful for graphing because it gives you direct insight into the line's angle and position.

The last step is to convert the equation from point-slope to slope-intercept form, which is \[ y = mx + b \]This compact form makes it easy to graph the line by identifying the slope \(m\) and the y-intercept \(b\), where the line crosses the y-axis.Starting from the previous point-slope equation:\[ y - 12 = -14(x - 5) \]We can rearrange it by distributing and simplifying:1. Distribute \(-14\) - \(y - 12 = -14x + 70\)2. Add 12 to both sides - \(y = -14x + 70 + 12\)3. Simplify further to find the equation - \(y = -14x + 82\)Now the equation is in the slope-intercept form where- The slope \(m\) is \(-14\),- The y-intercept \(b\) is \(82\), indicating the line crosses the y-axis at \((0, 82)\). This form is quick and easy to interpret if you need to represent or analyze the relationship visually or graphically.