91Ó°ÊÓ

To understand linear equations thoroughly, it's essential to get comfortable with the point-slope form. This form is particularly useful when you know a point on the line and the slope of that line.The equation for the point-slope form is written as:\[y - y_1 = m(x - x_1)\]Where:

• \((x_1, y_1)\) represents a specific point on the line.
• \(m\) is the slope of the line, a measure of its steepness.

This form is handy because it can help you write the equation of a line when you have limited information, just one point, and the slope.
In our exercise, knowing that the slope \(m\) is 3 and a point on the line is \((-4, 6)\), equips us to substitute directly into the point-slope form, resulting in the equation: \[y - 6 = 3(x + 4)\]From here, it becomes easier to plug in other points to see how they satisfy the equation or to solve for unknown values.

The slope of a line is a crucial concept in understanding linear equations as it defines the line's steepness and direction. Mathematically, the slope \(m\) is calculated as the change in the \(y\)-coordinate divided by the change in the \(x\)-coordinate between two points:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]This ratio tells us how much \(y\) changes for a unit change in \(x\).
In this exercise, the line's slope is given as 3, meaning for every increase of 1 in \(x\), \(y\) increases by 3. This constant rate of change is what defines the linearity of such functions.
Understanding the slope helps us to determine the direction (positive or upward, negative or downward) the line moves as well as its inclination. A positive slope, like in our example, indicates an increasing line as you move from left to right.

Solving for variables is a fundamental skill in algebra, involving manipulating equations to isolate the variable of interest. Here, we are tasked with finding the unknown \(x\) in the line's equation. Begin with the established equation from point-slope form:\[y - 6 = 3(x + 4)\]Knowing another point on the line \((x, 2)\), substitute \(y = 2\):\[2 - 6 = 3(x + 4)\]This simplifies to:\[-4 = 3(x + 4)\]Next, expand and arrange the equation:\[-4 = 3x + 12\]To solve for \(x\), eliminate the constant by subtracting 12 from both sides:\[-16 = 3x\]Finally, divide by 3 to isolate \(x\), leading to:\[x = -\frac{16}{3}\]This process of substituting, simplifying, expanding, and solving helps in understanding how variables interact in linear equations.