Problem 12 - College Algebra

Question

Use the given conditions to write an equation for each line in point-slope form and general form. Passing through \((5,-9)\) and perpendicular to the line whose equation is \(x+7 y-12=0\)

Step-by-Step Solution

Answer

The equation of the line in point-slope form is \(y = 7x - 44\), and in general form is \(-7x + y + 44 = 0\).

Step 1: Convert to Slope-Intercept Form

Convert the equation of the given line to slope-intercept form by isolating y. The given equation is \(x + 7y - 12 = 0\). Solve for y to get \(y = -\frac{1}{7}x + \frac{12}{7}\). So, the slope of the given line is -1/7.

Step 2: Find slope of Perpendicular Line

Since the slope of the original line is -1/7, the slope of the line perpendicular to it will be the negative reciprocal of -1/7, which is 7.

Step 3: Find Point-Slope Form

The point-slope form formula is \((y - y_1) = m(x - x_1)\), where \(m\) is the slope of the line and \((x_1, y_1)\) is a point on the line. Given that the line passes through the point (5, -9) and has a slope of 7, plug these values into the equation to get \((y - (-9)) = 7(x - 5)\), or \(y + 9 = 7x - 35\). Simplify to get \(y = 7x - 44\).

Step 4: Convert to General Form

Now, convert the equation from step 3 into general form, which is typically written as \(Ax + By + C = 0\), where A, B, C are integers and A should be positive. Convert the equation \(y = 7x - 44\) to get \(-7x + y + 44 = 0.\)

Key Concepts

Perpendicular Lines

When two lines are perpendicular to each other, they intersect at a right angle (90 degrees). One important characteristic of perpendicular lines involves their slopes. If you have the equation of a line, say its slope is \(m\), a perpendicular line will have a slope that is the negative reciprocal of \(m\).
This means if the slope \(m\) is a fraction \(-\frac{1}{7}\), the slope of any line perpendicular to it will be 7. Understanding perpendicular lines is crucial because the way their slopes are related helps in constructing equations of new lines from given lines.
To visually grasp this concept, if you imagine one line as flat and horizontal, a perpendicular line would be like an upright pole forming a cross with it.

Slope-Intercept Form

The slope-intercept form of a line is an algebraic way of expressing a linear equation. It is written as \(y = mx + b\), where \(m\) represents the slope of the line and \(b\) is the y-intercept. The y-intercept is the point where the line crosses the y-axis.
When you convert a linear equation to slope-intercept form, you can easily identify both the slope and the y-intercept. In this format, understanding the orientation and tilt of the line becomes more intuitive.
For example, for the equation obtained from a perpendicular line through the point (5, -9), it showed as \(y = 7x - 44\). Here, 7 is the slope, indicating a steep and positive rise, while -44 gives the point where our line crosses the y-axis.

General Form of Linear Equation

The general form of a linear equation is another way to express a line's equation, typically represented as \(Ax + By + C = 0\). Unlike the slope-intercept form, the focus here is not just on slope and intercept but rather on organizing the equation into a standard form used in various algebraic calculations.
One of the key points with this form is that \(A\), \(B\), and \(C\) are integers, making it versatile for solving systems of equations. The goal is to arrange the equation such that all terms involving variables are on one side and constant terms on the other side.
In our exercise, converting from slope-intercept to general form led to \(-7x + y + 44 = 0\). This format can facilitate use in broader contexts, like graphing, identifying intercepts, or integrating into further algebraic operations.