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Chapter 1: Problem 80

Write equations of the lines through the given point (a) parallel to and (b) perpendicular to the given line. $$6 x+2 y=9, \quad(-3.9,-1.4)$$

Short Answer

Expert verified
The equation of the line parallel to the given line is \(y = -3x - 12.3\) and the equation of the line perpendicular to the given line is \(y = (1/3)x - 0.5\).

Step by step solution

01

Find the slope of the given line

The given line is in standard form \(Ax + By = C\). So, rewrite it in slope-intercept form \(y = mx + b\) to easily find its slope. Divide all terms of the equation \(6x + 2y = 9\) by 2 to get \(3x + y = 4.5\). Rearranging, we get \(y = -3x + 4.5\). The coefficient of \(x\) is the slope of the line. Hence, slope of the given line \(\(m_1\) = -3.
02

Write the equation of the line parallel to the given line

Since parallel lines have equal slopes, the slope for our new line, \(\(m_2\)\), is also -3. Using the point-slope form of a line, where the given point is \((-3.9, -1.4)\) and \(\(m_2 = -3\)\), we have \(y - (-1.4) = -3(x - (-3.9))\). Simplifying this gives us the equation of the parallel line: \(y = -3x - 12.3\).
03

Write the equation of the line perpendicular to the given line

For perpendicular lines, the slopes are negative reciprocals. So the slope of our line \(\(m_3\)\) is the negative reciprocal of \(-3\), which is \(1/3\). Using the point-slope form of a line, where the given point is \((-3.9, -1.4)\) and \(\(m_3 = 1/3\)\), we have \(y - (-1.4) = (1/3)(x - (-3.9))\). Simplifying this gives us the equation of the perpendicular line: \(y = (1/3)x -0.5\).

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Most popular questions from this chapter

Graph each of the functions with a graphing utility. Determine whether the function is even, odd, or neither. $$\begin{aligned}&\begin{array}{ll}f(x)=x^{2}-x^{4} & g(x)=2 x^{3}+1 \\\h(x)=x^{5}-2 x^{3}+x & j(x)=2-x^{6}-x^{8}\end{array}\\\&k(x)=x^{5}-2 x^{4}+x-2 \quad p(x)=x^{9}+3 x^{5}-x^{3}+x \end{aligned}$$

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