Problem 9 - Algebra 1: Concepts and Skills

Question

Write the equation of the line passing through the point and perpendicular to the given line. $$ (5,2), y=-\frac{1}{2} x+4 $$

Step-by-Step Solution

Answer

The equation of the line passing through the point (5,2) and perpendicular to the given line is \(y = 2x - 8\).

Step 1: Find the Slope of the New Line

The slope of the given line is -1/2. The slope of a line perpendicular to another is the negative reciprocal of the original line's slope. So, the slope (\(m\')) of the new line is the negative reciprocal of -1/2, which is 2.

Step 2: Substitute the Point into the Line Equation

Substitute the given point (5,2) and the discovered slope (\(m\')) into the equation of a straight line (\(y = mx + c\)). We then get: 2 = 2*5 + c. Solve for c to find the y-intercept of the new line.

Step 3: Solve for c in the Equation

Solving for c, we get: c = 2 - 2*5 = 2 - 10 = -8. So, the y-intercept (\(c\')) of the new line is -8.

Step 4: Form the Equation of the New Line

With the slope (\(m\')) and y-intercept (\(c\')) of the new line, plug these values back into the equation of a straight line to get the equation for the new line. Hence the equation of the line is: \(y = 2x - 8\)

Key Concepts

Slope

The slope is a fundamental concept in geometry that describes the steepness or incline of a line. It is calculated as the ratio of the vertical change to the horizontal change between two points on a line. In simpler terms, it tells us how much the line rises or falls as we move along it.

A line with a positive slope goes upwards from left to right, whereas a line with a negative slope goes downwards. The formula for finding the slope (\( m \)) from two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

In the context of perpendicular lines, an interesting property arises: the slope of a line perpendicular to another is the negative reciprocal of the original line's slope. This means if you have a line with a slope of -\(\frac{1}{2}\), a line perpendicular to it will have a slope of 2. The negative reciprocal is found by flipping the fraction and changing the sign.

Equation of a Line

The equation of a line gives us a rule that describes the positions of points on a line, primarily in the Cartesian plane. One of the most commonly used forms of the equation of a line is the slope-intercept form, expressed as:

• \( y = mx + c \)

Here, \(m\) is the slope, and \(c\) is the y-intercept, which is where the line crosses the y-axis.

To write the equation of a line, you need to know its slope and one point it passes through. If a line is required to pass through a known point such as (5, 2), and you have determined that its slope is \(m = 2\), you substitute these values into the slope-intercept form to find \(c\).

After substituting the known values, you solve for \(c\) to complete the equation. This makes it easy to graph the line or understand its behavior.

Y-Intercept

The y-intercept of a line is the point where the line crosses the y-axis. This point provides valuable information about the line's position in a graph.

The y-intercept is represented by the 'c' in the slope-intercept equation, \( y = mx + c \). It tells us what the value of \(y\) is when \(x\) is zero. Finding the y-intercept is straightforward once you have the equation of the line. When you substitute other known values, you can isolate the y-intercept through simple algebra.

For example, once you inserted a point \((5, 2)\) into the equation of the line and worked through the calculations, you found the y-intercept to be \(-8\). This informs you that when \(x = 0\), \(y = -8\), which is where the line touches the y-axis. Understanding the y-intercept helps in quickly plotting a point on a graph, allowing for a clear visualization of the line's trajectory.