Problem 29 - College Algebra

Question

Find the equation of the line using the point-slope formula. Write all the final equations using the slope-intercept form. perpendicular to \(3 y=x-4\) and passes through the point \((-2,1)\) .

Step-by-Step Solution

Answer

The equation of the line is \(y = -3x - 5\).

Step 1: Find the slope of the given line

The equation of the line given is \(3y = x - 4\). First, let's rewrite it in the slope-intercept form, \(y = mx + b\), where \(m\) is the slope. Dividing the entire equation by 3 gives \(y = \frac{1}{3}x - \frac{4}{3}\). Therefore, the slope \(m\) of this line is \(\frac{1}{3}\).

Step 2: Determine the slope of the perpendicular line

Lines that are perpendicular to each other have slopes that are negative reciprocals. The slope of the perpendicular line will be the negative reciprocal of \(\frac{1}{3}\). The negative reciprocal is \(-3\).

Step 3: Use point-slope formula

The point-slope formula is \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is a point on the line and \(m\) is the slope. Substituting \(m = -3\) and the point \((-2, 1)\) into the formula gives \(y - 1 = -3(x + 2)\).

Step 4: Simplify to slope-intercept form

Expand and simplify the equation to convert it into the slope-intercept form \(y = mx + b\). Start by distributing \(-3\) in the equation: \(y - 1 = -3x - 6\). Adding 1 to both sides results in \(y = -3x - 5\), which is the equation in slope-intercept form.

Key Concepts

slope-intercept form

The slope-intercept form is a widely used way to express the equation of a straight line. It is written as \(y = mx + b\), where \(m\) represents the slope, and \(b\) is the y-intercept. This form is especially helpful because it clearly shows both the direction and steepness of the line (via the slope) and where the line crosses the y-axis (via the y-intercept).
Understanding this form is crucial in graphing linear equations quickly, as you can:

• Identify the slope \(m\), which tells you how much \(y\) changes for each unit increase in \(x\).
• Locate the point \((0, b)\), which marks where the line intersects the y-axis.

For example, in the line equation \(y = -3x - 5\), the slope is \(-3\), indicating the line falls 3 units in \(y\) for every 1 unit increase in \(x\). The y-intercept is \(-5\), meaning the line crosses the y-axis at \(y = -5\).
Using the slope-intercept form streamlines understanding and graphing a linear equation significantly.

negative reciprocals

Negative reciprocals are a key concept when dealing with perpendicular lines in geometry. If two lines are perpendicular, their slopes are negative reciprocals of each other. This means if one line has a slope \(m\), the perpendicular line will have a slope of \(-\frac{1}{m}\).
Here's how it works in practice:

• If the original slope is \(\frac{1}{3}\), the negative reciprocal would be obtained by flipping the fraction and changing the sign, giving \(-3\).
• This transformation is essential when you want to find the equation of a line perpendicular to a given line, as it directly impacts the slope in the slope-intercept form \(y = mx + b\).

Thus, knowing how to find negative reciprocals simplifies the process of determining the orientation of perpendicular lines.

equation of a line

The equation of a line can be determined in various forms, but one popular method is using the point-slope form. This is written as \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is a known point on the line and \(m\) is the slope.
The point-slope form is particularly useful when you know:

• The slope of the line \(m\).
• A specific point that the line passes through.

For instance, with a slope of \(-3\) and a point \((-2, 1)\), you can substitute these values into the point-slope formula:
\(y - 1 = -3(x + 2)\).
To find the slope-intercept form from here, simply simplify the expression: distribute the slope \(-3\), then solve for \(y\):
\(y - 1 = -3x - 6\) becomes \(y = -3x - 5\) after adding 1 to both sides. This gives the equation of the line in slope-intercept form.
Understanding these relationships helps in solving many problems concerning linear equations.