Problem 34 - Algebra 1: Concepts and Skills

Question

Write in slope-intercept form the equation of the line passing through the given point and perpendicular to the given line. $$ (0,0), y=-\frac{1}{4} x-7 $$

Step-by-Step Solution

Answer

The equation of the line that is perpendicular to \(y = -1/4x - 7\) and passes through the point (0,0) is \(y = 4x\).

Step 1: Find the slope of our line

From the given line \(y = -1/4 x - 7\), the slope is -1/4. Since our line is perpendicular to it, the slope of our line is the negative reciprocal of -1/4, which is 4.

Step 2: Find the y-intercept of our line

Since our line passes through the origin, (0,0), this means that the value of y when x is 0 is 0. Hence, the y-intercept of our line is 0.

Step 3: Write out the equation of the line

Now that we have the slope and y-intercept, we can substitute these into the slope-intercept formula ( \(y = mx + c\) ) to get the equation of our line. Substituting gives \(y = 4x + 0\), or more simply, \( y = 4x\)

Key Concepts

Perpendicular Line Equation

Understanding how to find the equation of a line that is perpendicular to a given line is crucial in geometry and coordinate algebra. Two lines are perpendicular if they intersect to form right angles with each other. The key to finding a perpendicular line equation is using the concept of slopes.

The slope of a line is a measure of its steepness and is calculated by the rise over run. For two lines to be perpendicular in a plane, the product of their slopes must be -1. This relationship is because the slopes of perpendicular lines are negative reciprocals of each other. If the slope of one line is 'm', then the slope of the line perpendicular to it will be '-1/m'.

In our example, the original line's slope is \( -\frac{1}{4} \) so the slope of the perpendicular line is the negative reciprocal, which is 4. So, the slope component in your perpendicular line equation would be 4, and using this, you can proceed to write the entire equation.

Negative Reciprocal Slope

When two lines are perpendicular to each other, their slopes exhibit a very specific relationship: they are negative reciprocals of one another. But what does 'negative reciprocal' mean?

A reciprocal of a number 'a' is simply '1/a' and its negative reciprocal is '-1/a'. If your line has a positive slope, say 'm', then the perpendicular line will have a slope '-1/m'. Conversely, if your original line has a negative slope, like \( -\frac{1}{4} \) in our problem, then you will take the reciprocal of \( \frac{1}{4} \) and change the sign, resulting in a positive 4.

It's important to note that the negative reciprocal of a vertical line, which would have an undefined slope, is a horizontal line with a slope of 0, and similarly, the negative reciprocal of a horizontal line is a vertical line.

Writing Linear Equations

Writing linear equations is a fundamental skill in algebra. A linear equation creates a straight line when graphed, and its most common form is the slope-intercept form, which is written as \(y = mx + c\). Here, 'm' represents the slope of the line, and 'c' stands for the y-intercept, the point where the line crosses the y-axis.

To write the equation of a line, you need to identify these two components. You have already learned how to find the slope of a perpendicular line through the negative reciprocal slope. For the y-intercept, you look for the point where your line crosses the y-axis (\(x=0\)).

Finding the Equation from a Point and a Slope

When you have a specific point and a slope (as in the example exercise), you can use the 'point-slope form' to create the equation, which is \( y - y_1 = m(x - x_1) \) where \( (x_1, y_1) \) is your point. You can then rearrange this equation to the slope-intercept form.

Y-intercept

The y-intercept of a line is fairly straightforward: it's the place where the line crosses the y-axis. It's a reliable starting point for graphing linear equations and aids in understanding the geometry of linear functions.

In standard slope-intercept form (\( y = mx + c \)), 'c' denotes the y-intercept. When you send \(x\) to zero, \(y\) will be equal to 'c'. Therefore, in the context of a straight line passing through the origin, like in our example, the y-intercept is 0 because the line crosses the y-axis at point (0,0).

Importance of the Y-intercept

The y-intercept is essential for graphing because it gives a point from which you can use the slope to find other points on the line. In our exercise, with the y-intercept being 0, plotting the line on a graph would start from the origin, making the process simpler.