Problem 20 - Algebra 2

Question

Write an equation in slope-intercept form for the line that satisfies each set of conditions. passes through \((2,-5),\) perpendicular to the graph of \(y=\frac{1}{4} x+7\)

Step-by-Step Solution

Answer

The equation is \(y = -4x + 3\).

Step 1: Understand Perpendicular Slope

The given line equation is in slope-intercept form: \(y = \frac{1}{4}x + 7\). The slope \(m\) of this line is \(\frac{1}{4}\). A line perpendicular to another has a slope that is the negative reciprocal of the original slope. Therefore, the slope of the line we need is \(-4\).

Step 2: Use Point-Slope Form

We have a point through which our line passes, \((2, -5)\), and we have determined the slope, \(-4\). We can use the point-slope form of a line equation: \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is the given point and \(m\) is the slope. Substituting, we have: \(y - (-5) = -4(x - 2)\).

Step 3: Simplify to Slope-Intercept Form

To convert the equation \(y + 5 = -4(x - 2)\) into slope-intercept form, distribute and simplify: \(y + 5 = -4x + 8\). Then, solve for \(y\): \(y = -4x + 8 - 5\), leading to the final equation \(y = -4x + 3\).

Key Concepts

Perpendicular Lines

When two lines meet at a right angle, or 90 degrees, they are considered to be perpendicular. One of the core characteristics of perpendicular lines is the relationship between their slopes. If you have the slope of one line, you can find the slope of a line perpendicular to it by using the concept of negative reciprocal.

This is a handy property that simplifies finding the right perpendicular path through a given point, making it useful in various geometric and algebraic applications. Perpendicular lines have crucial applications in creating accurate designs and understanding geometric shapes.

Slope

The slope of a line is a measure of its steepness and is represented typically by the letter \(m\). It is calculated as the change in \(y\) over the change in \(x\), or "rise over run." In the slope-intercept form \(y = mx + b\), the \(m\) represents the slope, while the \(b\) represents the \(y\)-intercept—the point where the line crosses the \(y\)-axis.

Understanding the slope is crucial, as it dictates the direction and angle of a line.

• A positive slope means the line inclines upwards to the right.
• A negative slope means it declines downwards to the right.
• A zero slope represents a horizontal line.
• An undefined slope corresponds to a vertical line.

Mastering the concept of slope is essential for graphing lines and solving many algebraic equations.

Point-Slope Form

The point-slope form is a way to represent a linear equation using a point and a slope. This form is especially valuable when you know a specific point on the line and the slope. The formula is expressed 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.

This form is practical for quickly building a line equation from real-world data or exercises where you have initial conditions like a particular point and slope. By substituting the values of your point and slope into this formula, you can handle complex algebraic tasks with ease, efficiently leading to the desired linear equation.

Negative Reciprocal

When discussing the slopes of perpendicular lines, the concept of negative reciprocal is pivotal. The negative reciprocal of a slope \(m\) is found by first taking the reciprocal of the slope \(\frac{1}{m}\) and then changing its sign. For instance, if a line has a slope of \(\frac{1}{4}\), the slope of a line perpendicular to it would be \(-4\).

This relationship is valuable when you need to generate a perpendicular line quickly, especially in algebra and geometry problems. Understanding and applying negative reciprocal will enable you to solve these problems efficiently, ensuring your lines are correctly aligned.