Problem 12 - 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. $$ (2,-10), m=-2 $$

Step-by-Step Solution

Answer

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

Step 1: Point-Slope Form of the Line

To find the equation of a line, start by using the point-slope form of a line, which is given by the formula: \[ y - y_1 = m(x - x_1) \] where \( (x_1, y_1) \) is the given point and \( m \) is the slope. Plug in the given point \((2, -10)\) and slope \( m = -2 \): \[ y + 10 = -2(x - 2) \]

Step 2: Simplify the Equation

Simplify the expression from Step 1: \[ y + 10 = -2x + 4 \] Subtract 10 from both sides to isolate \( y \): \[ y = -2x + 4 - 10 \] \[ y = -2x - 6 \]

Step 3: Convert to Standard Form

Now convert the equation to the standard form \( Ax + By = C \). Start from the equation: \[ y = -2x - 6 \] Add \( 2x \) to both sides to move \( x \) to the left side: \[ 2x + y = -6 \] At this point, the equation is in standard form, where \( A = 2 \), \( B = 1 \), and \( C = -6 \).

Key Concepts

Point-Slope Form

In the journey of understanding linear equations, the point-slope form is a pivotal concept. This form touches the very heart of how lines are represented mathematically when a point and a slope are known. The point-slope form of a line is expressed as \( y - y_1 = m(x - x_1) \), where:

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

Using these, you can easily determine the equation of a line. Let's consider our example, where the point \((2, -10)\) and the slope \(m = -2\) are given. By substituting these into the formula, we form the equation \( y + 10 = -2(x - 2) \). This setup is quite useful, especially when you need to express lines through different points with a known slope.
A good practice is simplifying this equation, revealing more intuitive insights about the line's direction and position. Once you get the structure from point-slope form, adjusting it to other forms becomes straightforward.

Standard Form of a Line

The standard form of a line is another classic representation used widely in mathematics. This form is represented as \( Ax + By = C \), with the condition that \(A\), \(B\), and \(C\) are integers, and \(A\) must be non-negative if possible. Converting equations into standard form is highly beneficial because it provides a clear and concise way to see the relationship between \(x\) and \(y\) variables.
Let's take the equation we derived from the point-slope form, \( y = -2x - 6 \), and transform it. By rearranging terms—specifically, adding \( 2x \) to both sides—we reach \( 2x + y = -6 \). This equation now fits the structure of the standard form perfectly, where \(A = 2\), \(B = 1\), and \(C = -6\).

• This versatile form aids in presenting a unified structure for linear equations.
• It is useful in systems of equations where multiple lines are analyzed simultaneously.

By mastering this format, you unlock the potential to compare lines and work effectively with systems of equations.

Slope-Intercept Form

The slope-intercept form is perhaps the most intuitive to students first learning about lines. It emphasizes the slope and y-intercept directly, showcasing the equation as \( y = mx + b \), where:

• \(m\) is the slope, indicating how steep the line is.
• \(b\) is the y-intercept, showing where the line crosses the y-axis.

In our earlier example, we simplified from the point-slope form to \( y = -2x - 6 \). Here, \(m = -2\) directly reveals the steepness of the line, while \(b = -6\) tells us the line crosses the y-axis at -6.
What's great about this form is its simplicity and directness in understanding the nature of a line. Each part of the equation clearly contributes to interpreting the line’s behavior and position on a graph. Whether plotting the line onto a graph or foreseeing how changes in \(m\) or \(b\) impact the graph's shape, this form is incredibly useful for visual learners and those getting comfortable with graphing.