Problem 5 - Elementary Algebra

Question

For Problems 1-12, find the equation of the line that contains the given point and has the given slope. Express equations in the form \(A x+B y=C\), where \(A, B\), and \(C\) are integers. $$ (-4,8), m=-\frac{1}{3} $$

Step-by-Step Solution

Answer

The equation of the line is \(x + 3y = 20\).

Step 1: Identify the Point-Slope Form

We start with the point-slope form of the equation of a line: \[ y - y_1 = m(x - x_1) \] where \((x_1, y_1)\) is the given point (-4, 8) and \(m = -\frac{1}{3}\) is the slope.

Step 2: Substitute the Point and Slope into the Equation

Substitute the given point \((-4, 8)\) and the slope \(m = -\frac{1}{3}\) into the point-slope form:\[ y - 8 = -\frac{1}{3}(x + 4) \]

Step 3: Simplify the Equation

Distribute the slope on the right-hand side:\[ y - 8 = -\frac{1}{3}x - \frac{4}{3} \]

Step 4: Move Terms to Form an Equation in Ax + By = C

To remove fractions, multiply the entire equation by 3 to eliminate the fraction:\[ 3(y - 8) = -x - 4 \]Simplify and rearrange to the standard form:\[ 3y - 24 = -x - 4 \]

Step 5: Rearrange to Standard Form

Move all terms to one side to get the standard form:\[ x + 3y = 20 \]This simplifies to the standard form of a linear equation with integer coefficients.

Key Concepts

Point-Slope Form

The point-slope form is a convenient way to find the equation of a line when you know a point on the line and the slope of the line. The general formula for the point-slope form is \[ y - y_1 = m(x - x_1) \]Here,

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

The beauty of the point-slope form is its direct connection to the slope and the specific point you have on the line. In this form, you're highlighting the rate at which the line rises or falls, which is what the slope represents.
For example, consider the point (-4, 8) and the slope \(-\frac{1}{3}\). By plugging these values into the point-slope form, you have:\[ y - 8 = -\frac{1}{3}(x + 4) \]This simple substitution quickly gets you started on forming the line's equation. From here, you can rearrange the equation into other forms, like the slope-intercept form or standard form, depending on your needs.

Standard Form

The standard form of a linear equation is expressed as \[ Ax + By = C \]where \(A\), \(B\), and \(C\) are integers. This form is useful because it provides a clear, structured way to present linear equations, allowing for easy comparison between different lines and straightforward solving of systems of equations.

• \(A\) and \(B\) should not both be zero.
• Typically, \(A\) is a non-negative integer, meaning \(A\) is positive or zero.

To convert an equation from point-slope form to standard form, you might need to clear fractions and rearrange terms. Starting from the simplified point-slope form:\[ y - 8 = -\frac{1}{3}x - \frac{4}{3} \]Multiply through by 3 to eliminate fractions:\[ 3(y - 8) = -x - 4 \]Simplifying:\[ x + 3y = 20 \]You've reached standard form, which is clean and integer-friendly. Equations in this form are optimal for not just graphing lines but also for computational purposes, like solving problems involving multiple linear equations.

Slope-Intercept Form

The slope-intercept form of a linear equation is one of the most widely used forms due to its simplicity and clarity. It is given by:\[ y = mx + b \]where

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

To understand this form, think of a line that starts at some point on the y-axis (the intercept) and rises or falls with a slope \(m\). The slope-intercept form is practical for graphing because you can quickly identify the starting point \((0, b)\) and draw the line using the slope. From the equation \[ y - 8 = -\frac{1}{3}(x + 4) \]if you simplify to solve for \(y\), you can express this line in slope-intercept form. After distributing and moving constants around, you would have something resembling:\[ y = -\frac{1}{3}x + \text{some number} \]While exercises involving finding the standard form tend not to focus on this, understanding the slope-intercept format provides deeper insight into how changes in \(m\) or \(b\) affect the line's graph. It's like the stepping stone for fully analyzing the behavior of linear functions.