Problem 24 - Algebra 1: Concepts and Skills

Question

Write in standard form an equation of the line that passes through the given point and has the given slope. \((-6,-7), m=-1\)

Step-by-Step Solution

Answer

The standard form equation of the line passing through the point (-6,-7) with slope of -1 is \(x + y = -13\).

Step 1: Using the point-slope form

Firstly, the given data (\(-6,-7)\) and \(m=-1\) are placed into the point-slope form, where \(x1=-6\), \(y1=-7\), and \(m=-1\). As a result, we get \(y-(-7)= -1(x - (-6))\).

Step 2: Simplifying the equation

Then, we simplify the equation. \(y+7=-1(x+6)\). Enable to distribute the negative into the term \((x+6)\) yielding the equation \(y+7=-x-6\).

Step 3: Transform to standard form

To achieve the standard form, roll the terms on one side of the equation. This gives the equation in the standard form as \(x + y= -13\).

Key Concepts

Point-Slope Form

The point-slope form of a linear equation is particularly useful when you have a point on the line and the slope of the line. The general format for this is:

• \( y - y_1 = m(x - x_1) \)

Here, \( (x_1, y_1) \) represents the coordinates of the given point, and \( m \) is the slope of the line. When you substitute these values into the formula, you can set up a basic equation of a line through any point with a defined slope.
Let's break this down in our example:

• The given point is \((-6, -7)\), so \(x_1 = -6\) and \(y_1 = -7\).
• The slope \( m \) is \(-1\).

Plug these into the point-slope equation and you have: \( y + 7 = -1(x + 6) \).This form allows you to explore the relationship between the point and the direction of the line.In summary, the point-slope form is a stepping stone in deriving other forms of line equations, like the slope-intercept and standard forms. This makes it valuable in understanding the nature and the path of the line in question.

Slope-Intercept Form

Another familiar approach to representing the equation of a line is the slope-intercept form. It's the form many students use instinctively because it provides both the slope and the y-intercept at a glance. The general formula is:

• \( y = mx + b \)

Where:

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

This form is quite intuitive as it allows you to quickly see how steep the line is and where it starts along the y-axis.For our example, after simplifying the point-slope form, you could further manipulate it into:

• \( y = -x - 13 \)

This expresses the line clearly by showing its slope \(-1\) and its y-intercept \(-13\).The slope-intercept form serves as an excellent way to graph linear equations and better understand the impact of changes in slope and y-intercept on the direction and position of the line.

Equations of Lines

Equations of lines are the foundation of linear algebra and geometry. Different forms of line equations serve different purposes during problem-solving. Understanding all these forms enables versatility when approaching linear equations.First, there is the **standard form** (the destination in our solution), represented as:

• \( Ax + By = C \)

In this form, the line's equation consolidates constants and variables to one side, making calculations with multiple linear equations more straightforward.In addition to the earlier discussed point-slope and slope-intercept forms, these three equations allow flexibility:

• **Point-Slope Form**: Helps in forming an equation when a point and slope are known.
• **Slope-Intercept Form**: Useful in quickly identifying slope and starting position (y-intercept).
• **Standard Form**: Great for solving systems of equations and eliminating fractions for a "clean" equation.

By transforming our equation from one form to another, you enhance algebraic skill sets like simplifying expressions and understanding inter-relations between various linear representations.The ability to switch smoothly between these forms can simplify tasks, such as determining intersections or analyzing two-variable datasets, ensuring that equations of lines remain a powerful tool in mathematics.