Is the line y = 3x + 7 parallel or perpendicular to the line 3x + 9y = 9?

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To determine whether the line y = 3x + 7 is parallel or perpendicular to the line 3x + 9y = 9, we first need to put both equations into the slope-intercept form, which is y = mx + b, where m indicates the slope of the line.

Step 1: Identify the slope of the first line

The line is already given in slope-intercept form:

y = 3x + 7

From this, we can see that the slope m_1 of the first line is:

m_1 = 3

Step 2: Rewrite the second equation in slope-intercept form

The second line is given as:

3x + 9y = 9

To convert this equation, we need to isolate y:

  1. Subtract 3x from both sides:

    2. 9y = -3x + 9

  2. Next, divide every term by 9 to solve for y:

    3. y = - rac{1}{3}x + 1

From this equation, we find that the slope m_2 of the second line is:

m_2 = - rac{1}{3}

Step 3: Determine the relationship between the slopes

Two lines are parallel if their slopes are equal, and they are perpendicular if the product of their slopes is -1. Let's calculate:

m_1 imes m_2 = 3 imes - rac{1}{3} = -1

Since the product of the slopes is -1, we can conclude that the lines are perpendicular.

Conclusion

The line y = 3x + 7 is perpendicular to the line 3x + 9y = 9.

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