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The slope of a line is a measure of how steep the line is and indicates the direction in which the line is moving. When you're trying to find the equation of a line, calculating the slope is a crucial first step. To start, remember that the slope formula is m = (y_2 - y_1) / (x_2 - x_1), where (x_1, y_1) and (x_2, y_2) are coordinates of two distinct points on the line.

For example, if you have the points \(1, 5\) and \(3, 9\), you subtract the y-coordinates of the second point from the first, and the x-coordinates of the second from the first, to get the numerical value of the slope. The calculation will look like this: \[m = (9 - 5) / (3 - 1) = 4 / 2 = 2\]. The slope here is 2, which means that for every unit the line moves horizontally, it moves 2 units vertically.

The y-intercept is another key element when writing the equation of a line. It is the point where the line crosses the y-axis. In the equation of a line \(y = mx + b\), b represents the y-intercept. After calculating the slope, you will use one of the given points alongside the slope to find the y-intercept.

Using the same points from the previous example, choose one point to substitute into the equation along with the slope. If \(1, 5\) is the chosen point and the slope \(m\) is 2, the equation now looks like \[5 = 2*1 + b\]. To solve for \(b\), subtract 2 from both sides to get \(3\). Thus, the y-intercept is 3. This tells us that when the line crosses the y-axis, the x-coordinate is 0, and the y-coordinate is 3.

Linear equations are algebraic expressions that represent straight lines when graphed on a coordinate plane. The standard form for a linear equation is \(y = mx + b\), where m stands for the slope, and b is the y-intercept. The slope determines the tilt of the line, and the y-intercept specifies where the line touches the y-axis.

With the slope and y-intercept calculated from the previous points given, \(2\) and \(3\) respectively, you can now write the linear equation. It will be in the form of: \[y = 2x + 3\]. This equation represents a line that goes through the points \(1, 5\) and \(3, 9\), slopes upward to the right, and intersects the y-axis at 3.