Problem 25 - 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. $$ m=-8 ; \quad(-1,-5) $$

Step-by-Step Solution

Answer

The equation in standard form is \( 8x + y = -13 \).

Step 1: Understand the Given Information

We are provided with the slope \( m = -8 \) and a point \( (-1, -5) \) that the line passes through. We need to find the equation of this line in the standard form \( Ax + By = C \).

Step 2: Recall the Point-Slope Form

The point-slope form of a line's equation is \( y - y_1 = m(x - x_1) \), where \((x_1, y_1)\) is a point on the line, and \(m\) is the slope. In this problem, \(x_1 = -1\), \(y_1 = -5\), and \(m = -8\).

Step 3: Substitute Values into Point-Slope Form

Plug the values into the point-slope form: \[ y + 5 = -8(x + 1) \]This is because \( y_1 = -5 \) and \( x_1 = -1 \).

Step 4: Simplify the Equation

Expand and simplify the equation:\[ y + 5 = -8x - 8 \]Subtract 5 from both sides to isolate \(y\):\[ y = -8x - 13 \]

Step 5: Convert to Standard Form

Rearrange the equation to the standard form \(Ax + By = C\). Add \(8x\) to both sides:\[ 8x + y = -13 \]Thus, the equation in standard form is: \( 8x + y = -13 \).

Key Concepts

Point-Slope Form

When we know a point on the line and the slope, the point-slope form is a valuable tool for finding the equation of that line.
This form is written as:

• \(y - y_1 = m(x - x_1)\)

Here, \((x_1, y_1)\) represents a specific point on the line, and \(m\) is the slope. By substituting the slope and the coordinates of the known point into this equation, we can derive a line equation.
This forms the basis for deriving other forms of line equations too.
Once you have substituted the given values into the point-slope format, you can easily convert the equation to other forms like the slope-intercept or standard form. For instance, in our exercise, by substituting \(-8\) as the slope and \((-1, -5)\) as the point, we derived the equation \(y + 5 = -8(x + 1)\).
This step helps us transition into more familiar formats for many students.

Standard Form Equation

The standard form of a linear equation is arranged as \(Ax + By = C\), where \(A\), \(B\), and \(C\) are integers, and it is common practice to keep \(A\) positive.
This format is preferred in many applications because it presents the equation without fractions and clearly displays both variables on one side.
Converting from the point-slope form into the standard form often involves rearranging terms and simplifying.
In our provided solution, we rearranged to obtain \(8x + y = -13\), ensuring all variable terms were on one side of the equation.
By having the result in standard form, it becomes easier to perform operations like addition or subtraction of equations, especially helpful in systems of equations.

Slope-Intercept Form

The slope-intercept form is perhaps one of the most intuitive for visualizing a line, expressed as \(y = mx + b\). Here, \(m\) denotes the slope, a measure of the line's steepness, and \(b\) represents the y-intercept, the point where the line crosses the y-axis.
This format clearly delineates the rate of change and the starting value, making it easy to graph and interpret linear relationships.
Once we have simplified an equation from the point-slope form, converting it into the slope-intercept form is straightforward.

• In our exercise, after simplification, we found \(y = -8x - 13\), directly illustrating the slope of \(-8\) and the y-intercept of \(-13\).

This form is exceptionally useful for quickly sketching a graph or understanding the direct relationship between \(x\) and \(y\).