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Chapter 2: Problem 58

Consider the following linear equations. Without graphing them, answer the questions below. a) \(y=x\) \(\quad\) b) \(y=-5 x+4\) \(\quad\) c) \(y=\frac{2}{3} x+1\) \(\quad\) d) \(y=-0.1 x+6\) \(\quad\) e) \(y=3 x-5\) \(\quad\) f) \(y=-x-1\) \(\quad\) g) \(2 x-3 y=6\) \(\quad\) h) \(6 x+3 y=9\) Which has the steepest slope?

Short Answer

Expert verified
Equation (e) has the steepest slope with a slope of 3.

Step by step solution

01

Identify the Slope

For each equation, identify the slope. The slope is the coefficient of the variable x in the form of y = mx + c.
02

Convert Equations to Slope-Intercept Form

Convert equations g) and h) to the slope-intercept form y = mx + c. This involves isolating y on one side of the equation.
03

Compare Slopes

Compare the absolute values of the slopes from each equation to determine which one is the steepest.
04

Conclusion

Identify the equation with the largest absolute slope value as the one with the steepest 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 one of the most common ways to express a linear equation. The general format of the slope-intercept form is \( y = mx + c \), where:
  • \( y \) is the dependent variable,
  • \( x \) is the independent variable,
  • \( m \) is the slope of the line, and
  • \( c \) is the y-intercept.
This format makes it straightforward to identify the slope and the y-intercept.The slope, \( m \), indicates the steepness and direction of the line: a positive \( m \) means the line is rising, while a negative \( m \) means it is falling.The y-intercept, \( c \), is where the line crosses the y-axis.
slope comparison
When comparing the steepness of different linear equations, you need to look at the slopes,\( m \). The larger the absolute value of the slope, the steeper the line.For example, in the equations
  • \( y = x \), slope = 1,
  • \( y = -5 x + 4 \), slope = -5,
  • \( y = \frac{2}{3} x + 1 \), slope = \( \frac{2}{3} \).
Regardless of the signs, you compare the numerical values of the slopes.A slope of -5 is steeper than \( \frac{2}{3} \) even though it's negative.So it's the size of the value that matters, not whether it's positive or negative.
absolute value of slope
The absolute value of the slope (\( m \)) represents the steepness of the line without considering the direction.It's important to use the absolute value when comparing slopes to ensure you're comparing the steepness correctly.For example:
  • If the slope of equation (a) is \( m = 2 \), the absolute value is \( |2| = 2 \).
  • If the slope of equation (b) is \( m = -3 \), the absolute value is \( |-3| = 3 \).
  • Between these, equation (b) has the steeper slope because 3 > 2.

This method is vital when determining which linear equation represents the steepest slope, regardless of the sign of \( m \).
isolating variables
To convert any linear equation to slope-intercept form, you must isolate the variable \( y \). Let's take examples from equations (g) and (h):
  • Equation (g): \( 2x - 3y = 6 \)
    To isolate \( y \),
    • First, move \( 2x \) to the other side: \( -3y = -2x + 6 \).
    • Then, divide every term by -3: \( y = \frac{-2}{-3}x + \frac{6}{-3} \).
    • This simplifies to \( y = \frac{2}{3}x - 2 \).
  • Equation (h): \( 6x + 3y = 9 \)
    To isolate \( y \),
    • Move \( 6x \) to the other side: \( 3y = -6x + 9 \).
    • Then, divide every term by 3: \( y = \frac{-6}{3}x + \frac{9}{3} \).
    • This simplifies to \( y = -2x + 3 \).
Using these steps helps in converting standard form equations into the slope-intercept form so you can easily identify the slope and y-intercept.

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Most popular questions from this chapter

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