Problem 12 - Elementary Algebra

Question

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. (Objective 1a) $$(2,-10), m=-2$$

Step-by-Step Solution

Answer

The equation is \(2x + y = -6\).

Step 1: Understand the Problem

We need to find the equation of a line that passes through the point \((2, -10)\) and has a slope of \(m = -2\). We will start by using the point-slope form of a line equation.

Step 2: Use Point-Slope Form

The point-slope form of a line is given by the equation \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is a point on the line and \(m\) is the slope. Substitute \((2, -10)\) for \((x_1, y_1)\) and \(-2\) for \(m\). This gives:\[y + 10 = -2(x - 2)\]

Step 3: Simplify the Equation

Distribute the slope on the right side and move terms:\[y + 10 = -2x + 4\]Subtract \(10\) from both sides to simplify the equation:\[y = -2x + 4 - 10\]\[y = -2x - 6\]

Step 4: Convert to Standard Form

Rearrange the equation \(y = -2x - 6\) to the form \(Ax + By = C\). Start by adding \(2x\) to both sides.\[2x + y = -6\]

Step 5: Ensure Integers in the Equation

The equation, \(2x + y = -6\), is already in the form \(Ax + By = C\), where \(A = 2\), \(B = 1\), and \(C = -6\), and all are integers.

Key Concepts

Point-Slope Form

The point-slope form of a linear equation is a useful tool for quickly writing the equation of a line when you know one point on the line and the slope. The formula is expressed as: \( y - y_1 = m(x - x_1) \), where

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

By substituting the known values into this equation, you can define a specific line. Think of the point-slope form as a starting place that easily adapts to changes. For example, consider a point \((2, -10)\) and a slope \(m = -2\). Plugging these into the formula gives you \( y + 10 = -2(x - 2) \). This intermediate step is incredibly handy for deriving the linear equation in different forms. The point-slope form, though not always the final version of an equation you'd use, provides a helpful bridge to other equation forms, such as the standard form.

Slope

The slope of a line measures its steepness and direction. It is often denoted by the letter \( m \) and can be calculated from two points \((x_1, y_1)\) and \((x_2, y_2)\) using the formula:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]When given directly, slope simplifies defining the behavior of a line. A positive slope rises from left to right, while a negative slope falls. A larger absolute value indicates a steeper line. For our example, a slope of \(-2\) means that for every unit you move to the right along the x-axis, the line moves down 2 units. The understanding of slope helps in visually predicting and plotting points on your line. Additionally, remember that if the slope is zero, the line is horizontal, and if undefined, it's vertical. Recognizing the slope connects naturally to using the point-slope form or transforming your equation into the standard form.

Standard Form of a Line

The standard form of a linear equation is expressed as \( Ax + By = C \). In this form, \(A\), \(B\), and \(C\) are integers, and it presents the equation in a straightforward style often most recognizable.To convert an equation like \( y = -2x - 6 \) (previously simplified from the point-slope form) to the standard form, rearrange terms:

• Add \(2x\) to both sides to get \( 2x + y = -6 \)

In the standard form, the x and y coefficients are neatly arranged on one side, making it easy to quickly determine intercepts and draw the line on a coordinate plane. This form also helps when solving systems of equations. For many applications, particularly when integers are required, the standard form gives us a clean and organized way to present a linear function. This makes it especially helpful in exams or standard computation tasks.