Problem 30 - Algebra A Combined Function

Question

Find an equation of each line with the given slope that passes through the given point. Write the equation in the form $A x+B y=C. $$ m=-\frac{1}{5} ; \quad(4,0) $$

Step-by-Step Solution

Answer

The equation of the line is \(x + 5y = 4\).

Step 1: Understand the Point-Slope Equation of a Line

The point-slope form for the equation of a line is given by \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \( (x_1, y_1) \) is a point on the line. Given the slope \( m = -\frac{1}{5} \) and point \( (4, 0) \), we will use these values in this equation.

Step 2: Plug Values into the Point-Slope Formula

Substitute the slope \( m = -\frac{1}{5} \) and point \( (4, 0) \) into the point-slope equation: \[ y - 0 = -\frac{1}{5}(x - 4). \] This simplifies to \( y = -\frac{1}{5}x + \frac{4}{5} \).

Step 3: Convert to Standard Form

The standard form of a linear equation is \( Ax + By = C \). We need to rearrange \( y = -\frac{1}{5}x + \frac{4}{5} \) to fit this form. First, eliminate the fraction by multiplying every term by 5: \[ 5y = -x + 4. \] Rearrange to get \( x + 5y = 4 \). This is the equation in standard form.

Key Concepts

Point-Slope Form

The point-slope form of a linear equation is useful when you know one point on the line and the slope of the line. This form is expressed as \( y - y_1 = m(x - x_1) \), where \( m \) stands for the slope and \((x_1, y_1)\) is a particular point on the line. This gives you a straightforward method to plug in known values and calculate the equation of any line easily.
For instance, if you're given a slope \( m = -\frac{1}{5} \) and a point \( (4, 0) \), you can substitute these values directly into the equation. This substitution results in:

• \( y - 0 = -\frac{1}{5}(x - 4) \)
• \( y = -\frac{1}{5}x + \frac{4}{5} \)

After plugging in these values, you arrive at the equation in the point-slope form, making it easy to transition to other forms if needed.

Standard Form of a Linear Equation

The standard form of a linear equation is represented as \( Ax + By = C \), where \( A \), \( B \), and \( C \) are integers, and \( A \) should ideally be a positive integer. It is a versatile format and is often used for ease of analysis or when working with integers.
From a given slope-intercept form \( y = -\frac{1}{5}x + \frac{4}{5} \), converting to standard form involves some algebraic manipulation. Begin by eliminating any fractions by multiplying the entire equation by the denominator, 5, in this case:

• \( 5y = -x + 4 \)

Rearrange this to get the equation into the form \( Ax + By = C \):

• \( x + 5y = 4 \)

This forms a neat equation in which all coefficients are integers, paving the way for straightforward graphical analysis and arithmetic operations.

Slope-Intercept Form

The slope-intercept form of a line is one of the most intuitive ways to write the equation of a line. It is expressed as \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept, meaning the point where the line crosses the y-axis. This form is particularly useful for graphing and quickly identifying the characteristics of the line such as direction and steepness.
Once we used the point-slope form to get \( y = -\frac{1}{5}x + \frac{4}{5} \), we have automatically arrived at the slope-intercept form. Here, you can see:

• The slope \( m \) is \(-\frac{1}{5}\)
• The y-intercept \( b \) is \(\frac{4}{5}\)

Identifying these elements can help in quickly sketching the graph of the line and understanding its behavior in a coordinate plane. This form also facilitates easy conversions to other equations like the standard form, when required.