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The first step in writing an equation for a line is to calculate the slope. The slope tells us how steep the line is and the direction it goes. We represent it with the letter \( m \). When you have two points, like \((4, 2)\) and \((-8, -16)\), you can use the slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] It's important to correctly choose which point is \((x_1, y_1)\) and which is \((x_2, y_2)\)—although the result will be the same regardless of the order, consistent labeling helps avoid mistakes. For the given points, this becomes: \[ m = \frac{-16 - 2}{-8 - 4} = \frac{-18}{-12} = \frac{3}{2} \]

• Change in y (rise) is \(-16 - 2 = -18\)
• Change in x (run) is \(-8 - 4 = -12\)

Always simplify the fraction to its lowest terms to get the slope. Here, the fraction \(-\frac{18}{12}\) simplifies to \(\frac{3}{2}\). So, the slope of our line is \(\frac{3}{2}\). That means for every 2 units you move right, the line moves up 3 units.

When you have the slope of a line and a point, it's easy to write the equation in point-slope form. The point-slope form of an equation is \[ y - y_1 = m(x - x_1) \] This formula is handy because you can plug in the slope \( m \) and any point \((x_1, y_1)\) on the line. Let's take the slope \( \frac{3}{2} \) and the point \((4, 2)\). Plug them into the formula: \[ y - 2 = \frac{3}{2}(x - 4) \] This equation represents the line and is particularly useful if you want to find other points on the line or transform it to another form.

• Choose which point to use; both given points will work.
• The choice of point affects the specific numbers in the equation but not the line it represents.

Point-slope form shows how the y-value of a point changes with the x-value, starting from a known point \((x_1, y_1)\). It's straightforward once you have the slope.

A linear equation in slope-intercept form is one of the most common ways to express a line. The slope-intercept form is \[ y = mx + b \] where \( m \) is the slope and \( b \) is the y-intercept, the point where the line crosses the y-axis. To convert from point-slope form to slope-intercept form: Start with the point-slope equation: \[ y - 2 = \frac{3}{2}(x - 4) \] Expand the equation: \[ y - 2 = \frac{3}{2}x - 6 \] Then solve for \( y \) by adding 2 to both sides, giving us: \[ y = \frac{3}{2}x - 4 \]

• \( y = mx + b \) form is straightforward and easy to interpret.
• \( b \) is where the line hits the y-axis, at y = -4 in this case.

This form not only tells us the slope and y-intercept but also makes graphing and analyzing the line simple. You can easily see how changes in \( x \) affect the line's \( y \) position and understand the line's overall trend.