Problem 91 - Algebra for College Students

Question

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 \(x+3 y=4\).

Step-by-Step Solution

Answer

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

Step 1 - 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 \(y = mx + b\).

Step 2 - 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 \(-\frac{1}{3}\).

Step 3 - 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 \(3\).

Step 4 - 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 \(m = 3\):\[y - (-3) = 3(x - (-4))\]\[y + 3 = 3(x + 4)\]

Step 5 - Simplify the equation

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

Step 6 - 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:\[3x - y = 9\]

Key Concepts

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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