Problem 8 - Algebra 1

Question

Write an equation of the line in point-slope form that passes through the given point and has the given slope. $$ (-5,-7), m=-2 $$

Step-by-Step Solution

Answer

The equation of the line in point-slope form that passes through the point (-5, -7) and has the slope -2 is \(y + 7 = -2(x + 5)\)

Step 1: Identify and substitute

Identify variables in point-slope form equation \(y - y_1 = m(x- x_1)\) which are provided in the question. With necessary values for the form as the slope \(m=-2\) and the point \((-5,-7)\), the equation will be formed by replacing these variables respectively i.e. \(m \) with \(-2\), \(x_1\) with \(-5\) and \(y_1\) with \(-7\).

Step 2: Implementing into equation form

Now implement the identified variables from the last step into point-slope form equation. Thus the equation becomes \(y - (-7) = -2(x - (-5)) \).

Step 3: Simplify the equation

Simplify the equation formed in step 2, after substituting the values into the equation. The equation will thus change to \(y + 7 = -2(x + 5)\).

Key Concepts

Algebraic Equations

Algebraic equations are mathematical statements that demonstrate the equality of two expressions. They are composed of variables, constants, and arithmetic operations such as addition, subtraction, multiplication, and division. Solving an algebraic equation typically involves finding the value(s) of the variable(s) that make the equation true.

For example, in the point-slope form equation of a line, which is an algebraic equation, we are looking for the linear relationship between the x and y coordinates on a graph. The point-slope form is expressed as \(y - y_1 = m(x - x_1)\), where \(m\) is the slope of the line and \((x_1, y_1)\) is a point on the line. By substituting known values into this algebraic framework, we can find the specific equation that represents our line.

Linear Equations

Linear equations form the backbone of linear algebra and are equations that create a straight line when graphed on a coordinate plane. These equations are always in the first degree, meaning neither the variable x nor y is raised to a power other than one.

The most common forms of linear equations are the slope-intercept form, \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept, and the point-slope form, which we focus on here. The beauty of the point-slope form is its ability to quickly generate the equation of a line when you know one point it passes through and its slope. To solidify the concept, recall our problem which yields the point-slope equation \(y + 7 = -2(x + 5)\) after plugging the known values into the point-slope template. This represents the linear relationship, graphable as a straight line, defined by those specific parameters.

Slope of a Line

The slope of a line is a measure of its steepness, usually denoted by the letter \(m\). It can be computed as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. The slope can be thought of as 'rise over run', represented by the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\).

Slope is a crucial concept when understanding linear equations. For example, in our exercise, the slope is given as \(-2\). This negative slope indicates that the line is inclined downwards from left to right. A positive slope would suggest an upward incline, while a slope of zero denotes a horizontal line. When the slope is undefined (due to a vertical line), we cannot use the usual slope-intercept or point-slope forms without some modification, as this situation entails a 'run' of zero, and division by zero is undefined in mathematics.