Problem 47 - Algebra 1: Concepts and Skills

Question

Write in standard form the equation of the line that passes through the given point and has the given slope. (Lesson 5.4 ) $$ (10,6), m=-2 $$

Step-by-Step Solution

Answer

The standard form of the equation of the line that passes through the point (10, 6) and has slope -2 is \(2x + y = 26\).

Step 1: Identify the given point and slope

The point given is (10, 6) and the slope \(m\) is -2.

Step 2: Use the slope-intercept form

The slope-intercept form of the line is \(y = mx + c\). Plug in the given point into this equation to find the y-intercept \(c\). That gives \(6 = -2*10 + c\), which simplifies to \(c = 26\). So, the equation of the line in slope-intercept form is \(y = -2x + 26\).

Step 3: Convert to standard form

To convert the slope-intercept form \(y = -2x + 26\) to standard form \(Ax + By = C\), move the terms on the left side to change the form of the equation: \(2x + y = 26\).

Key Concepts

Standard Form

The standard form of an equation of a line is expressed as \( Ax + By = C \), where \( A \), \( B \), and \( C \) are integers, and \( A \) and \( B \) are not both zero. This form is particularly useful because it provides a clear, consistent way to represent the equation of a line. Standard form makes it easy to identify key characteristics, like intercepts or parallelism among lines.
Standard form is often favored in mathematical situations that require integer coefficients or where simplifying intersections of lines is necessary. There are a few important rules to remember when using standard form:

• \( A \) should be a positive integer. If it isn’t, multiply the entire equation by -1.
• The equation should have no fractions or decimals; if present, multiply the equation by a common denominator to clear them.
• \( A \), \( B \), and \( C \) should be coprime; meaning they have no common factor other than one.

In our example, converting \( y = -2x + 26 \) to standard form gives us \( 2x + y = 26 \), which adheres to these rules and clearly shows the structure of the line.

Slope-Intercept Form

The slope-intercept form of a line's equation is one of the most straightforward and commonly used formats, expressed as \( y = mx + c \). Here, \( m \) represents the slope of the line, and \( c \) indicates the y-intercept where the line crosses the y-axis.
This form is particularly useful because it provides immediate insights:

• The slope \( m \) tells us how "steep" the line is, and in which direction it’s inclined.
• The y-intercept \( c \) makes it easy to graph the line, since one point is immediately known.

In our example, substituting the slope \( m = -2 \) and coordinates of the point \((10, 6)\) into the slope-intercept formula allows us to solve for \( c \):
\[ 6 = -2 \times 10 + c \]
Solving for \( c \), we find \( c = 26 \). Thus, the slope-intercept form of our line’s equation is \( y = -2x + 26 \).

Slope

The slope of a line is a measure of its inclination relative to the horizontal axis, quantifying the rate of change in the \( y \)-direction for a given change in the \( x \)-direction. Mathematically, it is defined as \( m = \frac{\Delta y}{\Delta x} \), which represents the rise over run.
Understanding the slope helps in determining whether a line climbs or falls as we move along the x-axis:

• A positive slope means the line rises as it moves from left to right.
• A negative slope indicates the line falls over the same motion.
• A slope of zero signifies a perfectly horizontal line.
• An undefined slope is characteristic of a vertical line.

In our example, the slope is given as \( m = -2 \), indicating a downward slope of 2 units for each unit increase in \( x \). This detail was used to find the line's equation in different forms. It tells us how the coordinates of the line incline or decline relative to each other.

Point-Slope Form

The point-slope form is another method to represent the equation of a line, expressed as \( y - y_1 = m(x - x_1) \). It’s particularly valuable when you are given a point on the line and the slope, a common scenario in geometrical problems.
Consider this an intermediate step between calculating basic slope and forming a full equation:

• \( (x_1, y_1) \) are the coordinates of the given point.
• \( m \) is the slope of the line.

For our example, substituting \( m = -2 \) and the coordinates \((10, 6)\) into the point-slope formula gives:
\[ y - 6 = -2(x - 10) \]
This equation provides a different perspective in establishing the line's equation, allowing for easy direction towards converting into other forms such as slope-intercept or standard form. The point-slope form is flexible and often simple for quickly deducing linear relationships from known points and slopes.