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Chapter 2: Problem 20

Answer each question about the properties of the given line(s). Which line has a steeper slope: \(y=5 x+10\) or the line passing through the points (-5,0) and (0,11)\(?\)

Short Answer

Expert verified
The line \(y = 5x + 10\) has a steeper slope.

Step by step solution

01

Understand the Slope-Intercept Form

The equation of a line in slope-intercept form is given by \(y = mx + b\), where \(m\) is the slope of the line and \(b\) is the y-intercept. For the line \(y = 5x + 10\), the slope \(m\) is \(5\).
02

Calculate the Slope of the Second Line

The slope \(m\) of a line that passes through two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]. Substituting the points (-5,0) and (0,11), we get: \[ m = \frac{11 - 0}{0 + 5} = \frac{11}{5}. \] Thus, the slope of the line is \(\frac{11}{5}\).
03

Compare the Slopes of Both Lines

The slopes being compared are \(5\) from the first line and \(\frac{11}{5}\) from the second line. To compare them, convert \(5\) to a fraction, which is \(\frac{25}{5}\). Since \(\frac{25}{5}\) is greater than \(\frac{11}{5}\), the first line has a steeper slope.

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Key Concepts

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

Slope-Intercept Form
The slope-intercept form is a way to write the equation of a straight line. The formula for this form is given by \(y = mx + b\). Here, \(m\) represents the slope, which indicates how steep the line is, and \(b\) denotes the y-intercept, the point where the line crosses the y-axis. Understanding this form helps us quickly identify the slope from an equation.Let's break it down:
  • \(y\) is the dependent variable and corresponds to vertical coordinates on the graph.
  • \(x\) is the independent variable that denotes horizontal coordinates.
  • \(m\) tells you how much \(y\) changes as \(x\) increases by one unit.
  • \(b\) is where the line touches the y-axis.
In our example, the line given by \(y = 5x + 10\) has a slope of \(5\), which means for every unit increase in \(x\), \(y\) increases by \(5\) units.
Comparing Slopes
When comparing slopes, we're essentially determining which line is steeper. A larger absolute value of the slope \(m\) means a steeper line. To compare slopes, make sure they're both in a similar form, like fractions.Here's how to compare:
  • Convert integer slopes to fractions with a denominator that's common to your other slope. For example, convert \(5\) to \(\frac{25}{5}\).
  • Compare the numerators if the denominators are the same. A larger numerator means a steeper line.
  • If one line has a slope of \(\frac{25}{5}\) and another has \(\frac{11}{5}\), the first line is steeper.
In our scenario, the slope of \(5\) is converted to \(\frac{25}{5}\), making it easy to compare with \(\frac{11}{5}\). Clearly, \(\frac{25}{5}\) is greater, indicating a steeper line.
Calculating Slope from Points
To calculate the slope \(m\) from two points, use the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). This formula helps find the rate of change between two points on a line. It measures how much \(y\) changes with each unit change in \(x\).Let's use this formula:
  • Identify your two points: \((x_1, y_1)\) and \((x_2, y_2)\).
  • Subtract the y-values: \(y_2 - y_1\).
  • Subtract the x-values: \(x_2 - x_1\).
  • Divide to find the slope: \(\frac{y_2 - y_1}{x_2 - x_1}\).
With the points \((-5, 0)\) and \((0, 11)\), plug in the values: \(\frac{11 - 0}{0 + 5} = \frac{11}{5}\). This gives the line's slope between these two points.

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Most popular questions from this chapter

Answer each question about the properties of the given line(s). Determine the \(x\) -intercept of the line \(y=4 x-8\)

In exercises \(16-18\), factor the given function, and graph the function. $$ b(x)=x^{3}+6 x^{2}+12 x+8 $$

In exercises \(5-11,\) write each function in the form \((x+a)^{2}+b\) and identify the values of \(a\) and \(b\). $$ p(x)=x^{2}+11 x+4 $$

Write each statement using inequalities. $$ x \in(5,6] \cup[7,8) $$

Write an equation for each line in the indicated form. Write the equation of the line in slope-intercept form passing through the points (-1,0) and (3,6) .
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