Problem 51 - Precalculus with Limits

Question

In Exercises 51-64, find the slope-intercept form of the equation of the line that passes through the given point and has the indicated slope \(m\). Sketch the line. \((0, -2)\), \(m = 3\)

Step-by-Step Solution

Answer

The slope-intercept form of the equation of the line that passes through the point (0, -2) and has the slope \(m=3\) is \(y=3x-2\).

Step 1: Identify the given values

The slope of the line \(m\) is given as 3 and the line passes through the point \((0, -2)\). So, we know that when \(x=0\), \(y=-2\).

Step 2: Substitute the known values into the slope intercept formula

Putting \(m=3\), \(x=0\), \(y=-2\) into the formula \(y=mx+b\), we get: \(-2=3(0)+b\).

Step 3: Solve for \(b\)

After substituting the values into the formula, we see that it simplifies to: \(-2=b\). This tells us that the y-intercept is -2.

Step 4: Write the final equation

Now that we have found the y-intercept, we place the slope and the y-intercept into the slope intercept form \(y=mx+b\). The final equation is \(y=3x-2\).

Key Concepts

Linear Equations

Linear equations are foundational for understanding algebra and are used to describe a straight line on a graph. They have the general form of y = mx + b, where m represents the slope or the steepness of the line, and b is the y-intercept, which is the point where the line crosses the y-axis. It's essential to grasp that every linear equation yields a straight line when graphed and each straight line can be represented by such an equation.

For instance, if you are given a point through which the line passes and its slope, you can determine the specific linear equation for that line. In the case of the exercise, we have the slope m = 3 and a point on the line (0, -2). This information is enough to calculate the y-intercept and find the precise equation of the line.

Slope of a Line

The slope of a line is a measure of its steepness and is usually represented by the letter m. Mathematically, slope is calculated as the 'rise' over the 'run' between any two points on the line; that is, the change in the y-values divided by the change in the x-values. The formula used is m = (y2 - y1) / (x2 - x1).

Positive slopes indicate that the line is tilted upwards as we move to the right, while negative slopes indicate a line tilted downwards. The greater the absolute value of the slope, the steeper the line. Zero slope means the line is horizontal, and an undefined slope (or infinite) indicates a vertical line. In our exercise, the given slope m is 3, which means for every 1 unit we go right, we go up by 3 units.

Y-Intercept

The y-intercept is a specific point where the line crosses the y-axis, and it is denoted as b in the slope-intercept form of a linear equation y = mx + b. Essentially, it is the value of y when x is 0. It provides a starting point for drawing a line on a graph and is key to understanding the position of the line in relation to the origin (0,0).

In the provided exercise, substituting the known values into the slope-intercept formula and solving for b yielded the y-intercept as -2. This tells us that the line crosses the y-axis below the origin, at the point (0, -2).

Graphing Lines

Graphing lines on a coordinate system involves plotting points and connecting them to reveal the overall shape and direction of the line. The slope-intercept form, y = mx + b, is particularly handy for graphing because it directly provides the slope and the y-intercept, which are two critical pieces of information needed to draw an accurate line.

To graph the line from the exercise, you start at the y-intercept (0, -2). From there, you use the slope, m = 3, to find another point. Since the slope is the rise over run, you move up 3 units and 1 unit to the right. Plot this second point, draw a line through both points, and extend it across the graph; this is the visual representation of your linear equation y = 3x - 2.