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Chapter 2: Problem 8

Which equation has the steepest slope? a. \(y=2-7 x\) b. \(y=2 x+7\) c. \(y=-2+7 x\)

Short Answer

Expert verified
Equations a and c both have the steepest slopes with an absolute value of 7.

Step by step solution

01

- Identify the slope in each equation

The slope of a linear equation in the form of \(y = mx + b\) is the coefficient \(m\). Identify the slope for each given equation:a. \(y=2-7x\): Here, the slope \(m = -7\).b. \(y=2x+7\): Here, the slope \(m = 2\).c. \(y=-2+7x\): Here, the slope \(m = 7\).
02

- Compare the slopes

Compare the absolute values of the slopes identified in Step 1 to determine which is the steepest:a. The slope is \(-7\), and its absolute value is \(7\).b. The slope is \(2\), and its absolute value is \(2\).c. The slope is \(7\), and its absolute value is \(7\).
03

- Determine the equation with the steepest slope

Since the absolute values of the slopes in equations a and c are both \(7\), these two equations have the steepest slopes. Therefore, both equations a and c have the steepest slopes.

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Key Concepts

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

Understanding Slope
In a linear equation, the slope is a measure of how steep the line is. It tells us how much the y-value (vertical) changes for a unit change in the x-value (horizontal). The slope is represented by the letter 'm' in the equation form \( y = mx + b \).
If the slope is positive, the line rises from left to right. If it is negative, the line falls from left to right. For example:
  • In equation \( y = 2x + 7 \), the slope \( m \) is 2, meaning the line rises by 2 units for every 1 unit it moves to the right.
  • In equation \( y = 2 - 7x \), the slope \( m \) is -7, meaning the line falls by 7 units for every 1 unit it moves to the right.
Understanding the slope helps us determine the steepness and direction of the line.
The Absolute Value of a Slope
The absolute value of the slope helps us compare steepness without caring whether the line rises or falls. The absolute value is a number's distance from zero and is always positive or zero.
For example, in the equation \( y = 2 - 7x \), even though the slope is -7, its absolute value is 7. This tells us that the steepness is 7, irrespective of the line's direction.
  • So, for slope -7, absolute value is 7.
  • For slope 2, absolute value is 2.
  • For slope 7, absolute value is 7.
By comparing absolute values, we ignore the negative sign and focus on steepness.
Comparing Slopes
To determine which equation has the steepest slope, compare the absolute values of their slopes. Steepness is how vertical a line is, regardless of its direction.
As in the exercise:
  • Equation \( y = 2 - 7x \) has a slope of -7. Absolute value = 7.
  • Equation \( y = 2x + 7 \) has a slope of 2. Absolute value = 2.
  • Equation \( y = -2 + 7x \) has a slope of 7. Absolute value = 7.
We see the absolute values 7 from both -7 and 7 are greatest. Hence, equations \( y=2-7x \) and \( y=-2+7x \) have the steepest slopes.
Forms of Linear Equations
Linear equations are usually written in the form \(y = mx + b\), where:
  • \(y\) is the dependent variable.
  • \(x\) is the independent variable.
  • \(m\) represents the slope.
  • \(b\) represents the y-intercept, where the line crosses the y-axis.
This form is called the slope-intercept form. Familiarizing yourself with this form makes it easy to identify the slope and y-intercept quickly. For example,
  • In \(y = 2 - 7x\), rewrite it as \(y = -7x + 2\) to identify the slope \(m = -7\) and y-intercept \(b = 2\).
  • In \(y = 2x + 7\), the slope \(m = 2\) and the y-intercept \(b = 7\).
Recognizing these forms helps solve and understand linear equations better.

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