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Chapter 2: Problem 72

Write an equation for each line. Then graph the line. \(m=2,\) through \((1,3)\)

Short Answer

Expert verified
The equation of the line is \(y = 2x + 1\). To graph it, start at (0,1) and use a slope of 2 to plot another point, then draw a line through these points.

Step by step solution

01

Write the Point-Slope Form of the Equation

Begin by using the point-slope formula: \(y - y_1 = m(x - x_1)\), where \(m\) is the slope and \((x_1, y_1)\) is the point the line passes through. Plug in the given slope \(m = 2\) and the coordinates of the given point \((1, 3)\) into the formula to get the equation of the line.
02

Simplify the Equation

Substitute the given values into the point-slope form to get \(y - 3 = 2(x - 1)\). Simplify the equation by distributing the slope and moving the constant term to the other side to get the equation in slope-intercept form, \(y = mx + b\).
03

Convert to Slope-Intercept Form

First, distribute the 2 on the right side to get \(y - 3 = 2x - 2\). Then add 3 to both sides to find the y-intercept (b): \(y = 2x + 1\).
04

Graph the Line

To graph the line with the equation \(y = 2x + 1\), start at the y-intercept (0,1) on the y-axis. Then, use the slope 2, which means go up 2 units for every 1 unit you go to the right, and plot a second point. Draw the line through these points.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Point-Slope Form
The point-slope form is particularly useful when you know one point on a line and its slope. It is written as \(y - y_1 = m(x - x_1)\), where \(m\) represents the slope and \(x_1, y_1\) are the coordinates of the known point. To create an equation from the point-slope form, simply plug in the slope and the coordinates.

For example, if a line has a slope of 2 and passes through the point (1,3), we substitute these values into the formula, yielding \(y - 3 = 2(x - 1)\). This is the first step in writing the linear equation for a line when the slope and a single point are known.
Slope-Intercept Form
The slope-intercept form is another popular way to express the equation of a line. It is written as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept, which is the point where the line crosses the y-axis. To convert the point-slope form to the slope-intercept form, distribute the slope to both \(x\) and \(x_1\) and solve for y to isolate the y-intercept on one side.

For the given line, after simplification, we get \(y = 2x + 1\), which clearly displays both the slope and the y-intercept. This form is highly useful for quickly graphing the line and understanding its behavior.
Graphing Linear Equations
Graphing linear equations involves plotting points on a coordinate system and connecting them to form a straight line. Using the slope-intercept form \(y = mx + b\) assists in this process. You begin by marking the y-intercept on the y-axis. Next, apply the slope \(m\), which is usually written as a ratio (rise over run).

For our example, the line equation \(y = 2x + 1\) tells us to start at the y-intercept (0,1) and then move up 2 units (rise) and 1 unit to the right (run) to find another point on the line. By connecting these points with a straight line, the graph of the equation is completed.
Linear Equations
Linear equations are the foundation of algebra and define lines in a coordinate system. They can take various forms, including point-slope and slope-intercept, but all linear equations ultimately describe a straight line. Equations of lines are constructed based on knowing either two points, a point and the slope, or the y-intercept and the slope.

In the context of the given exercise, the linear equation was constructed using a point and the slope, demonstrating the versatility and connectivity of the different forms of linear equations. Once the line's equation is determined, analyzing its graph reveals its behavior and intersection points with the axes or other lines.

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