Problem 121 - Precalculus Essentials

Question

Will help you prepare for the material covered in the next section. Write the slope-intercept form of the equation of the line passing through (-3,1) whose slope is the same as the line whose equation is \(y=2 x+1\)

Step-by-Step Solution

Answer

The slope-intercept form of the equation of the line that passes through the point (-3,1) and has the same slope as the line \(y=2x+1\) is \(y = 2x + 7\).

Step 1: Identify Slope of Given Line

Begin by identifying the slope of the line given in the problem \(y=2x+1\). The equation is given in slope-intercept form, so the slope (\(m\)) is the coefficient of \(x\), which is \(2\) in this case.

Step 2: Identify the Required Point

Next, identify the point (\(-3,1\)) through which our required line passes. In the coordinate system, this point will represent \(x_1\) and \(y_1\), respectively.

Step 3: Calculate the y-intercept

To find the y-intercept (\(b\)) of our required line, we'll use the complete form of the slope-intercept equation which is \(y - y_1 = m(x - x_1)\). We know the slope (\(m = 2\)), and we know the coordinates for the point (\(x_1=-3, y_1=1\)). Substituting these values into the formula, we'll solve for \(b\). That will result in \(b = y_1 - m * x_1 = 1 - 2*(-3) = 7\).

Step 4: Write the Line Equation

Once we have figured out the slope which is \(2\) and y-intercept \(b\) which is \(7\), these can be substituted back into the slope-intercept form equation to give the equation of the line: \(y = 2x + 7\).

Key Concepts

Understanding Linear Equations

Linear equations are fundamental in algebra and represent relationships between two variables with a constant rate of change. These equations can be visualized as straight lines when graphed on a coordinate plane. Generally speaking, a linear equation in two variables, such as 'x' and 'y', can be written in the form of ax + by = c, where 'a', 'b', and 'c' are constants. However, the slope-intercept form, which is y = mx + b, provides more immediate insights into the graph of the line, by handily expressing the slope (m) and the y-intercept (b) of the line.

In the slope-intercept form, 'm' represents the steepness or the incline of the line, and 'b' represents the point where the line crosses the y-axis. This makes it particularly useful for quickly graphing the line and understanding its behavior without extensive calculations. When given a linear equation, identifying these elements allows us to sketch the graph accurately and appreciate the linear relationship it conveys.

By using the slope-intercept form, it becomes possible to predict additional points on the line, understand the direction the line goes depending on the slope's sign, and determine how 'steep' the change is based on the slope's magnitude.

Graphing of Lines

Graphing lines is an important skill in mathematics, as it provides a visual representation of linear equations. The process of plotting a line involves several steps and requires an understanding of the coordinate plane, which is composed of a horizontal 'x-axis' and a vertical 'y-axis'. To graph a line given its equation in slope-intercept form, the first step is to identify the y-intercept (b) and plot that point directly on the y-axis.

Following this initial point, the slope (m) determines how we move to plot the next point. For instance, a slope of 2 means you rise 2 units vertically for every 1 unit you move to the right (the positive x direction). If the slope is negative, the line falls as it moves to the right. Repetition of this 'rise over run' using the slope ensures that all points lie on the straight line defined by the equation.

To illustrate a coherent picture of the line, connect the plotted points with a straight edge. This graphic interpretation of linear equations is essential for understanding concepts like the rate of change, direct variation, and patterns of linear relationships.

Equation of a Line

The equation of a line expresses the relationship between x and y coordinates on a graph. In slope-intercept form, y = mx + b, it succinctly represents the characteristics of the line, with 'm' indicating the line's slope and 'b' indicating where the line intersects the y-axis. There are other forms, like point-slope form y - y1 = m(x - x1), which is particularly useful when we know a point on the line (x1, y1) and its slope (m).

When forming the equation of a line, we can gather information from different starting points. If we know two points on the line, we can calculate the slope by dividing the change in y-coordinates by the change in x-coordinates. Then, with the slope and a point, we can either plug the values into the slope-intercept form and solve for 'b', or we can use the point-slope form directly to write the equation of the line. Understanding how to manipulate these forms and convert between them allows us to approach a variety of problems involving lines, such as predicting values, model relationships, and solving systems of equations involving multiple lines.