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Chapter 1: Problem 25

Just as the \(y\) -intercept of a line is the \(y\) value at which the line crosses the \(y\) -axis, the \(x\) -intercept is the \(x\) value at which the line crosses the \(x\) -axis. In Exercises \(22-25,\) find an equation of a line with the given \(x\) -intercept and slope. \(x\) -intercept \(3,\) no slope (Hint: If slope is \(\frac{\text { rise }}{\text { run }}\) , when would there be no slope?)

Short Answer

Expert verified
The equation of the line is \( x = 3 \).

Step by step solution

01

Understand the Slope

The exercise states that the line has no slope. In mathematical terms, a line with no slope is a vertical line.
02

Identify the X-Intercept

The given x-intercept is 3. This means the line crosses the x-axis at the point (3, 0).
03

Formulate the Equation of the Line

Since the line is vertical and passes through the x-intercept 3, its equation can be written as \[ x = 3 \] because a vertical line always has the equation of the form \[ x = a \], where \( a \) is the x-intercept.

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Key Concepts

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

Understanding the X-Intercept
In the context of a line on a graph, the x-intercept is the point where the line crosses the x-axis. This means that at the x-intercept, the value of y is always zero because the point lies directly on the x-axis. In general, any point on the x-axis can be represented as \( (x, 0) \). For example, if the x-intercept of a line is 3, it means the line crosses the x-axis at the point (3, 0). Understanding the concept of the x-intercept is crucial as it helps in formulating the equation of the line.
Understanding the Slope
The slope of a line measures its steepness and is determined by the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. Mathematically, the slope is given by \( \frac{\text{{rise}}}{\text{{run}}} \). However, in some cases, a line can have no slope. This happens when the line is vertical. A vertical line rises infinitely without running horizontally, making the run value zero. Because dividing by zero is undefined in mathematics, a vertical line is said to have no slope. Therefore, when you see a line with no slope, you instantly know it is vertical.
Understanding a Vertical Line
A vertical line is a special case in geometry where the line extends infinitely in the vertical direction without any horizontal movement. Therefore, the x-coordinate of every point on a vertical line is the same, while the y-coordinate can be any value. The equation of a vertical line is always of the form \( x = a \), where \( a \) is the x-intercept of the line. For example, if the x-intercept is 3, the equation of the vertical line will be \( x = 3 \). This means that no matter what the value of y is, the x-coordinate is always 3.

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Most popular questions from this chapter

Below is world population data for the years 1950 through 1990. $$\begin{array}{|c|c|}\hline & {\text { Population }} \\ {\text { Year }} & {\text { (billions) }} \\ \hline 1950 & {2.52} \\ {1960} & {3.02} \\ {1970} & {3.70} \\ {1980} & {4.45} \\ {1990} & {5.29} \\ \hline\end{array}$$ a. Plot the points on a graph with 鈥淵ears since 1900鈥� on the horizontal axis and 鈥淧opulation (billions)鈥� on the vertical axis. Try to fit a line to the data. b. Write an equation to fit your line. c. Use your equation to project the world population for the year 2010, which is 110 years after 1900. d. What does your equation tell you about world population in 1900? Does this make sense? Explain. e. According to United Nations figures, the world population in 1900 was 1.65 billion. The UN has predicted that world population in the year 2010 will be 6.79 billion. Are the 1900 data and the prediction for 2010 different from your predictions? How do you explain your answer?

A blue plane flies across the country at a constant rate of 400 miles per hour. a. Is the relationship between hours of flight and distance traveled linear? b. Write an equation and sketch a graph to show the relationship between distance and hours traveled for the blue plane. c. A smaller red plane starts off flying as fast as it can, at 400 miles per hour. As it travels it burns fuel and gets lighter. The more fuel it burns, the faster it flies. Will the relationship between hours of flight and distance traveled for the red plane be linear? Why or why not? d. On the axes from Part b, sketch a graph of what you think the relationship between distance and hours traveled for the red plane might look like.

Hoshi drew graphs for \(y=x\) and \(y=-x\) and noticed that the lines crossed at right angles at the point \((0,0) .\) Then he drew graphs for \(y=x+4\) and \(y=-x+4\) and noticed that the lines crossed at right angles again, this time at the point \((0,4) .\) He tried one more pair, \(y=x-4\) and \(y=-x-4 .\) Once again the lines crossed at right angles, at the point \((-4,0)\). (Table not Copy) Hoshi made this conjecture: 鈥淲hen you graph two linear equations and one has a slope that is the negative of the other, you always get a right angle.鈥� a. Do you agree with Hoshi鈥檚 conjecture? Why or why not? b. Draw several more pairs of lines that fit the conditions of Hoshi鈥檚 conjecture, with different slope values. Do your drawings prove or disprove Hoshi鈥檚 conjecture? c. If you think Hoshi鈥檚 conjecture is false, where do you think he made his mistake?

Evaluate each expression for \(a=2\) and \(b=3\). $$ a^{b}+b^{a} $$

Use the distributive property to rewrite each expression without using parentheses. \(p q\left(\frac{1}{p^{2}}-\frac{q}{p}\right)\)
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