Question

Find an equation of the line that passes through the point and has the indicated slope. Sketch the line. $$ (3,-2) \quad m=3 $$

Short answer

The equation of the line that passes through the point (3, -2) and has a slope of 3 is \(y = 3x - 11\).

Step-by-step solution

Insert the slope into the equation

Given the slope \(m = 3\), we can substitute \(m\) in the basic equation of a line \(y = mx + b\). It will result in \(y = 3x + b\).

Substituting the given point

We know that the line passes through the point (3,-2). So the x- and y-values of this point satisfy the equation of the line. The point (3,-2) implies \(x = 3\) and \(y = -2\). So, we then substitute these values into the equation to find \(b\). It will be \(-2 = 3*(3) + b\).

Calculate the y-intercept

Upon solving the last equation we find the y-intercept, \(b\). It will be \(-2 = 9 + b => b = -2 - 9 => b = -11\).

Plugging the y-intercept to get the final equation of the line

Now we can substitute \(b\), the y-intercept we found, back into the equation, to obtain the equation of the line: \(y = 3x - 11\).

Key concepts

Slope of a Line

Understanding the slope of a line is fundamental in algebra and geometry. The slope is a measure of the steepness or the incline of a line. It is often represented by the letter 'm' and calculated by dividing the change in the y-coordinate by the change in the x-coordinate between two distinct points on the line, which can be expressed as ewline ewline ewline ewline ewline ewline \( m = \frac{rise}{run} = \frac{\Delta y}{\Delta x} \).ewline ewline ewline ewline ewline ewline For the given exercise, the slope is provided as 3, indicating a relatively steep incline, and that for every unit increase in x, the y-value increases by 3 units.
ewline ewline ewline ewline ewline ewline Understanding the concept of slope is crucial since it tells us not only how lines rise and fall but also whether they are parallel (having the same slope) or perpendicular (having negative reciprocal slopes).

Y-Intercept

The y-intercept of a line is the point at which the line crosses the y-axis. It's another crucial element when one needs to understand lines as it gives an initial value from which the line starts. The y-intercept is defined by the coordinate (0,b), where 'b' is the value of the y-coordinate when x equals zero.
ewline ewline ewline ewline ewline ewline In our example, after substituting the given point (3, -2) and the slope 3 into the slope-intercept form equation \( y = mx + b \), we found that the y-intercept 'b' was -11. This means our line crosses the y-axis at the point (0, -11). One can consider the y-intercept as the starting point of the line when graphed on a coordinate plane.

Point-Slope Form

The point-slope form is an algebraic formula used to determine the equation of a line when given a point on the line and the slope. This form is essential when the y-intercept is not readily available. The point-slope formula is represented as \( y - y_1 = m(x - x_1) \), where \( m \) is the slope, and \( (x_1, y_1) \) are the coordinates of the given point through which the line passes.
ewline ewline ewline ewline ewline ewline For the problem provided, if we were to use the point-slope form instead of the slope-intercept form, we would plug in the given slope \( m = 3 \) and the coordinates of the point (3, -2) into the formula to get \( y - (-2) = 3(x - 3) \). After simplifying, we'd arrive at the same equation \( y = 3x - 11 \) that defines the line. This form is incredibly useful when working with linear equations as it is straightforward and adaptable to different types of problems.