Question

Line 1 is defined by \(y=3 x+7\), and line 2 is perpendicular to line 1. What is the slope of line 2 ? \(\bigcirc-3\) O \(\frac{1}{3}\) \(\bigcirc-\frac{1}{3}\) O I don't know.

Short answer

The slope of line 2 is \(-\frac{1}{3}\).

Step-by-step solution

Find the slope of the given line

The general form of a line equation is \(y = mx + b\), where \(m\) is the slope. Here, line 1 is given by the equation \(y = 3x + 7\). So, the slope (m1) of line 1 is 3.

Determine the slope of a line perpendicular to line 1

The slope (m2) of a line that is perpendicular to line 1 is the negative reciprocal of m1. The negative reciprocal of a number is found by flipping the number and changing its sign. So, the slope of line 2 would be \(-\frac{1}{m1}\), which equals \(-\frac{1}{3}\).

Key concepts

Slope-Intercept Form

Understanding the slope-intercept form is crucial for working with linear equations. The slope-intercept form of a linear equation is expressed as \(y = mx + b\). In this equation, \(m\) stands for the slope of the line, which tells you how steep the line is. The \(b\) represents the y-intercept, which is the point where the line crosses the y-axis.
This form is very useful because it provides clear information about the line's slope and intercept, allowing you to easily graph it or compare it with other lines. When dealing with problems involving linear equations, identify the slope and y-intercept to gain insights about the line's behavior.
For example, in the exercise, line 1 is given by the equation \(y = 3x + 7\). Here, the slope \(m\) is \(3\), indicating the line rises 3 units vertically for every 1 unit it moves horizontally.

Negative Reciprocal

The concept of negative reciprocals is key when dealing with perpendicular lines. When two lines are perpendicular, their slopes have a specific relationship. One line's slope is the negative reciprocal of the other.
To find the negative reciprocal, flip the fraction and change its sign. For instance, if a slope is \(3\), the negative reciprocal is \(-\frac{1}{3}\). This is because you flip \(3\) over to get \(\frac{1}{3}\), and then change its sign to negative.
Using this concept, if a line has a slope of \(3\), a line perpendicular to it will have a slope of \(-\frac{1}{3}\). Recognizing and using negative reciprocals helps in determining the slopes of lines that are perpendicular to each other.

Linear Equations

Linear equations describe straight lines and can be represented in several forms. The most common form is the slope-intercept form \(y = mx + b\), which we discussed earlier. Linear equations are called 'linear' because they graph as straight lines. Each solution to the equation represents a point on this line.
Linear equations might also appear in other forms like the standard form, but they all boil down to describing a line on a coordinate plane. The key is understanding the parts of the equation: the coefficient of \(x\) is the slope and it dictates the tilt or steepness of the line. The constant term is the y-intercept, showing where the line hits the y-axis.
In the exercise, grasping these parts of the equation \(y = 3x + 7\) was essential to finding the slope of the perpendicular line. Remember, linear equations are powerful and versatile tools for representing relationships between variables in a simple linear form.