In the equation √(2 + m²) = m + 3, the variable m represents the slope of a line. What is the slope of a line that is perpendicular to the line whose slope is m?
A. -6/7 B. -7/6 C. 2/9 D. 6/7 E. 7/6
(D) First, we need to solve for m. Note, that m must be greater or equal to -3, because square root must be a non-negative number. If we square both sides of the equation, to eliminate the square root, we'll get: 2 + m² = m² + 6m + 9. Then we proceed to solve for m: 2 + m² = m² + 6m + 9. 2 = 6m + 9 -7 = 6m m = -7/6. -7/6 > -3, so it is solution of the equation.
Now we need to understand that the slopes of perpendicular lines are negative reciprocals of each other.
Therefore, all lines that are perpendicular to a line with a slope of -7/6 have slopes of 6/7. The correct answer is choice (D).
Could you, please, explain why m is greater or equal to -3 and something regarding line slopes?
When we solve an equation like √(2 + m²) = m + 3 and find a value (or values) of m, we need to make sure everything is Ok with original statement. For that we can either plug in a value (or values) of m in it:
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