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Chapter 2: Problem 35

For each equation, find the slope. If the slope is undefined, state this. $$ 3 x=-15 $$

Short Answer

Expert verified
The slope is undefined.

Step by step solution

01

Simplify the equation

To find the slope, first simplify the equation if needed. Given the equation: \[ 3x = -15 \] divide both sides by 3 to isolate x.
02

Solve for x

Divide both sides by 3:\[ x = \frac{-15}{3} \]which simplifies to:\[ x = -5 \]This is now in the form of a vertical line, where x is constant.
03

Interpret the slope of a vertical line

For the equation \( x = -5 \), this line is vertical. Vertical lines have an undefined slope because their rise/run ratio is infinite or indeterminate.

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Key Concepts

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

vertical line
A vertical line represents a special case in coordinate geometry. Unlike other lines, a vertical line runs parallel to the y-axis. In simpler terms, it goes straight up and down. One key feature of a vertical line is that all points on the line have the same x-coordinate. For instance, the equation \( x = -5 \) means that every point on this line has an x-coordinate of -5.

Here are a few more points to understand vertical lines better:
  • Vertical lines are always parallel to the y-axis.
  • The form of a vertical line equation is always \( x = c \), where c is a constant.
  • Because all x-values are the same, there's no way to calculate a slope as there's no 'run' (change in x).
This ties directly into the concept of slope, which we'll delve into next.
undefined slope
Slope is a measure of how steep a line is. For most lines, it's calculated by the ratio of the 'rise' (change in y) to the 'run' (change in x). This is typically represented as \( m = \frac{y_2 - y_1}{x_2 - x_1} \).

However, for vertical lines, the 'run' is zero, as the x-coordinates do not change. You can't divide by zero in mathematics, so we say that the slope of a vertical line is undefined. In other words:
  • A vertical line means the slope formula results in division by zero.
  • This makes the slope undefined, often represented by the symbol \( \text{m is undefined} \).
The undefined slope is an important concept, particularly when learning about different types of lines in algebra.
linear equations
Linear equations are fundamental in algebra and coordinate geometry. These equations describe a straight line when plotted on a graph. Generally, a linear equation can be written in the form \( Ax + By = C \). Depending on the values of A and B, you can have:
  • Horizontal lines (when A=0)
  • Vertical lines (when B=0)
  • Slanted lines (when neither A nor B is zero)
In the case of our example, \( 3x = -15 \), we have a scenario where it's only in terms of x, representing a vertical line. These simple rules help understand and visualize linear equations much better.

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Most popular questions from this chapter

Solve each equation graphically. Then check your answer by solving the same equation algebraically. $$ x+2=3 $$

Determine whether each equation is linear. Find the slope of any nonvertical lines. $$ y=\frac{10}{x} $$

Solve each equation graphically. Then check your answer by solving the same equation algebraically. $$ 2 x+5=1 $$

Solve. If appropriate, classify the equation as either \(a\) contradiction or an identity. $$ 10-x=5(x+2) $$

Write equations for two functions \(f\) and \(g\) such that the domain of \(f+g\) is \(\\{x | x \text { is a real number and } x \neq-2 \text { and } x \neq 5\\}\). Answers may vary.
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