Problem 22 - 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. \((-2,7), m=-4\)

Step-by-Step Solution

Answer

So, the equation of the line in standard form is \(4x + y = -1\).

Step 1: Filling the point and slope in point-slope form

Substitute \((x1, y1) = (-2, 7)\) and \(m = -4\) into the point-slope equation \(y - y1 = m(x - x1)\). This will result to \(y - 7 = -4(x + 2)\).

Step 2: Distributing and simplifying the equation

Next, distribute \( -4 \) into the parentheses to simplify the equation: \(y - 7 = -4x - 8\).

Step 3: Turning into standard form

To convert this equation into standard form, the equation should look like this: \(Ax + By = C\). So, add \(4x\) and \(7\) to both sides to form: \(4x + y = -1\).

Key Concepts

Point-Slope Form

The point-slope form of a line is an algebraic equation that allows us to construct the equation of a line when we know one point on the line and the slope of that line. It is written as: \[ y - y_1 = m(x - x_1) \]
In this formula, the point \( (x_1, y_1) \) is a point that the line passes through, and \( m \) is the slope of the line.

To use this form effectively, first identify the given point and the slope. In our problem, the given point is \( (-2,7) \) and the slope is \(m=-4\). By substituting these values into the point-slope form, we get the equation \( y - 7 = -4(x + 2) \), which is the initial step of our exercise solution.

Why Point-Slope Form is Useful:

• Quickly provides the linear equation when a point and a slope are known.
• Allows for easy plotting of the line on a graph.
• Facilitates the understanding of how the slope affects the position of the line.

Standard Form of a Line

The standard form of a line's equation is another way to write the equation of a line. It is typically represented as: \[ Ax + By = C \]
where \( A \) and \( B \) are the coefficients of \( x \) and \( y \) respectively, and should be integers. The value \( C \) is the constant. In the standard form, \( A \) should be non-negative, and \( A \) and \( B \) should not both be zero.

To convert an equation from point-slope to standard form, one has to rearrange and simplify the equation by moving \( x \) and \( y \) terms to one side, and the constant to the other side. In the given exercise, after rearranging the point-slope form equation, we achieve \( 4x + y = -1 \), which is now in standard form.

Characteristics of Standard Form:

• The coefficients \( A \) and \( B \) should be whole numbers.
• Preferably, \( A \) is non-negative.
• Provides a clear view of the intercepts on the \( x \) and \( y \) axes.

Slope of a Line

The slope of a line is a measure of its steepness or gradient and is commonly represented by \( m \). It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line. Formally, it is written as: \[ m = \frac{{rise}}{{run}} = \frac{{y_2 - y_1}}{{x_2 - x_1}} \]
In the exercise, the slope is provided as \( m=-4 \), which indicates that for every unit of horizontal change to the right, there is a four-unit vertical change downwards (since it is negative). This negative slope results in a line that tilts downwards as it moves from left to right.

Understanding Slope:

• A positive slope indicates a line rising from left to right.
• A negative slope indicates a line falling from left to right.
• A slope of zero indicates a horizontal line.
• An undefined slope (division by zero) indicates a vertical line.

Getting comfortable with the concept of slope allows for better graphing and understanding of linear relationships in equations.