Problem 57 - Elementary Algebra

Question

Contains the point \((-1,3)\) and is perpendicular to the line \(2 x-y=4\)

Step-by-Step Solution

Answer

The equation of the line is \(x + 2y = 5\).

Step 1: Find the slope of the given line

The given line is represented by the equation \(2x - y = 4\). To find its slope, let's rearrange it into the slope-intercept form \(y = mx + b\). By solving for \(y\), we get \(y = 2x - 4\). Therefore, the slope of the given line, \(m_1\), is 2.

Step 2: Determine the perpendicular slope

The slopes of perpendicular lines are negative reciprocals. Therefore, if the slope of the given line is \(m_1 = 2\), then the slope \(m_2\) of the line that is perpendicular to it will be \(-\frac{1}{2}\).

Step 3: Write the equation of the perpendicular line

Use the point-slope form of the equation \(y - y_1 = m(x - x_1)\), where \(m\) is the slope and \((x_1, y_1)\) is a point on this line. The given point is \((-1, 3)\) and we found \(m = -\frac{1}{2}\). Apply these values to get: \[y - 3 = -\frac{1}{2}(x + 1)\].

Step 4: Simplify the equation

Distribute the slope on the right side: \[y - 3 = -\frac{1}{2}x - \frac{1}{2}\].Add 3 to both sides to solve for \(y\): \[y = -\frac{1}{2}x - \frac{1}{2} + 3\].Combine constants: \[y = -\frac{1}{2}x + \frac{5}{2}\].

Step 5: Express the equation in standard form

To convert the equation \(y = -\frac{1}{2}x + \frac{5}{2}\) to standard form, multiply through by 2 to eliminate fractions: \[2y = -x + 5\].Rearrange it to get \[x + 2y = 5\].

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is one of the most popular ways to express the equation of a line. It is written as:- \( y = mx + b \)
Where:

• \( m \) represents the slope of the line.
• \( b \) is the y-intercept, the point where the line crosses the y-axis.

To convert a linear equation into this form, you rearrange the terms so that \( y \) is by itself on one side. Once in this form, \( m \) and \( b \) can be easily identified, providing a straightforward way to graph the line.For example, given an equation like \( 2x - y = 4 \), converting to slope-intercept form involves isolating \( y \). The transformed equation is \( y = 2x - 4 \), where the slope \( m \) is 2, and the y-intercept \( b \) is \(-4\). This makes it easier to visualize and understand the line's behavior in relation to the coordinate plane.

Point-Slope Form

Point-slope form is incredibly useful when you know a point on the line and the line's slope. This linear equation form is written as:- \( y - y_1 = m(x - x_1) \)
Where:

• \( m \) is the slope of the line.
• \((x_1, y_1)\) is a known point on the line.

To use this form, you simply plug in the slope and the coordinates of the known point and solve for \( y \). This form is especially advantageous when dealing with perpendicular lines, as you can quickly compute the perpendicular slope as the negative reciprocal of the given line's slope.Consider our example, where the original line with slope \( 2 \) has a perpendicular slope of \(-\frac{1}{2}\). Given the point \((-1, 3)\), you substitute into the point-slope equation to get: \[ y - 3 = -\frac{1}{2}(x + 1) \]which lays the groundwork for finding the complete equation of the line.

Standard Form

The standard form is yet another way of expressing a linear equation. It is written as:- \( Ax + By = C \)
Where:

• \( A \), \( B \), and \( C \) are integers.
• \( A \) should be non-negative.
• When possible, \( A \), \( B \), and \( C \) should have no common divisors other than 1.

Transforming an equation to standard form involves moving terms around so that both \( x \) and \( y \) terms are on one side of the equation, with the constant on the opposite side.In our example, we transformed the slope-intercept equation \( y = -\frac{1}{2}x + \frac{5}{2} \) to standard form by clearing the fraction. Multiplying through by 2 gives \( 2y = -x + 5 \), and rearranging leads to: \[ x + 2y = 5 \]This form is especially helpful when dealing with systems of equations, as it makes the equations easier to solve jointly.