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Chapter 3: Problem 91

Find the equation of line l in each case and then write it in standard form with integral coefficients. Line \(l\) goes through \((-4,-3)\) and is perpendicular to \(x+3 y=4\).

Short Answer

Expert verified
The equation of line \(l\) in standard form is \(3x - y = 9\).

Step by step solution

01

- Identify the slope of the given line

First, we need to find the slope of the given line. The equation of the given line is in standard form: \[x + 3y = 4\]. To find its slope, convert it to the slope-intercept form \(y = mx + b\).
02

- Convert to slope-intercept form

Rewriting the equation \(x + 3y = 4\):\[3y = -x + 4\]\[y = -\frac{1}{3}x + \frac{4}{3}\]Thus, the slope \(m\) of this line is \(-\frac{1}{3}\).
03

- Find the slope of the perpendicular line

Since line \(l\) is perpendicular to the given line, its slope will be the negative reciprocal of \(-\frac{1}{3}\). The negative reciprocal of \(-\frac{1}{3}\) is \(3\). So, the slope of line \(l\) is \(3\).
04

- Use point-slope form

The equation of a line in point-slope form is \(y - y_1 = m(x - x_1)\). Using the point \((-4, -3)\) and slope \(m = 3\):\[y - (-3) = 3(x - (-4))\]\[y + 3 = 3(x + 4)\]
05

- Simplify the equation

Simplify the equation from point-slope form to slope-intercept form:\[y + 3 = 3x + 12\]Subtract 3 from both sides:\[y = 3x + 9\]
06

- Convert to standard form

To write the equation in standard form with integral coefficients, move all terms to one side of the equation:\[y - 3x = 9\]Multiply through by -1 to get positive coefficients for \(x\):\[-y + 3x = -9\]Thus, the equation in standard form is:\[3x - y = 9\]

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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 of a linear equation is one of the most common ways to express the equation of a line. It's represented as \(y = mx + b\). Here:
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Most popular questions from this chapter

Determine whether each pair of lines is parallel, perpendicular, or neither. $$y-3 x=5,3 x+y=7$$

Find an equation of the line that goes through the given point and has the given slope. Give the answer in slope-intercept form. See Example 5 (-3, -1) with slope 6

Let \(f(x)=3 x-2, g(x)=-x^{2}+3 x-2,\) and \(h(x)=|x+2|\). Evaluate each expression. See Example 9. $$ f(100) $$

Determine which of the ordered pairs \((1,3),(-2,5)\) \((-6,-4),\) and \((7,-8)\) satisfy each compound or absolute value inequality. $$|x-y|>2$$

Graph each pair of lines in the same coordinate system using the slope and y-intercept. $$ \begin{aligned} &y=\frac{3}{4} x+3\\\ &y=-\frac{4}{3} x+1 \end{aligned} $$
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