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Chapter 4: Problem 369

Use slopes and \(y\) -intercepts to determine if the lines are parallel. \(9 x-5 y=4 ; \quad 5 x+9 y=-1\)

Short Answer

Expert verified
The lines are not parallel.

Step by step solution

01

- Convert the first equation to slope-intercept form

Convert the first equation, which is given in standard form, to the slope-intercept form, which is \(y = mx + c\).Start with the equation: \(9x - 5y = 4\).Isolate y by solving for y in terms of x:\[-5y = -9x + 4\]Divide both sides by -5:\[y = \frac{9}{5}x - \frac{4}{5}\] The slope of the first line, \(m_1\), is \(\frac{9}{5}\).
02

- Convert the second equation to slope-intercept form

Convert the second equation to the slope-intercept form.Start with the equation: \(5x + 9y = -1\).Isolate y by solving for y in terms of x:\[9y = -5x - 1\]Divide both sides by 9:\[y = -\frac{5}{9}x - \frac{1}{9}\] The slope of the second line, \(m_2\), is \(-\frac{5}{9}\).
03

- Compare the slopes

To determine if the lines are parallel, compare the slopes of each line.The slope of the first line is \(\frac{9}{5}\) and the slope of the second line is \(-\frac{5}{9}\).Since \(\frac{9}{5} eq -\frac{5}{9}\), the slopes are not equal.Thus, the lines are not parallel.

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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 used to express a linear equation in a way that makes it easy to find the slope and the y-intercept of the line. The general form is given by:

y = mx + c,
where:
  • m = slope of the line (rise over run)
  • c = y-intercept (where the line crosses the y-axis)
To convert any linear equation into slope-intercept form, you need to isolate y on one side. In this exercise, the given equations were:

1. 9x - 5y = 4,
2. 5x + 9y = -1.
Converting each to slope-intercept form allows us to compare their slopes easily. Let鈥檚 see the detailed process starting with the first equation:

  • Start with 9x - 5y = 4. Move 9x to the right side: -5y = -9x + 4.
  • Divide all terms by -5 to isolate y: y = 9/5 * x - 4/5. Here, the slope (m) is 9/5.
Similarly, for the second equation:
  • Start with 5x + 9y = -1. Move 5x to the right: 9y = -5x - 1.
  • Divide all terms by 9: y = -5/9 * x - 1/9. Here, the slope (m) is -5/9.
Now that the slopes are in clear view, we can compare them!
Standard Form Conversion
Many linear equations are initially given in standard form:

Ax + By = C.
In this format, A, B, and C are integers. The process of converting from standard form to slope-intercept form involves rearranging the equation to solve for y. Here is the step-by-step method:

  • Move the x-term to the right side by subtracting or adding it to both sides.
  • Divide every term by the coefficient of y to isolate y completely.
Let鈥檚 take a look at the equations given in this exercise and how they were converted:

First Equation: 9x - 5y = 4
  • Move 9x to the right: -5y = -9x + 4.
  • Divide all terms by -5: y = 9/5 * x - 4/5.
Second Equation: 5x + 9y = -1
  • Move 5x to the right: 9y = -5x - 1.
  • Divide all terms by 9: y = -5/9 * x - 1/9.
Conversion like this simplifies comparisons of slope and y-intercept, making it easier to determine relationships between lines.
Comparing Slopes
Comparing the slopes of two lines helps determine their geometric relationship. If two lines have:
  • Equal slopes but different y-intercepts, they are parallel.
  • Different slopes, they intersect at some point and are neither parallel nor identical.
  • Equal slopes and equal y-intercepts, they are actually the same line.
In the exercise, after converting both equations to slope-intercept form, we have:

1. First Line Slope ( m1 ): 9/5,
2. Second Line Slope ( m2 ): -5/9.
Comparing these slopes:
Since 9/5 is not equal to -5/9 , the lines are not parallel.

Understanding the implications of slopes helps in identifying the relationship between lines:
  • Parallel lines never meet because they rise and run at the same rate.
  • Intersecting lines meet at a point because their slopes vary.
Thus, the given lines, having different slopes, are certainly not parallel.

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