Problem 56 - Algebra 1

Question

Write the point-slope form of the equation of the line that passes through the two points. $$ (4,-2),(-9,5) $$

Step-by-Step Solution

Answer

The point-slope form of the line that passes through the points (4,-2) and (-9,5) is \(y + 2 = -\frac{7}{13}(x - 4)\)

Step 1: Find the slope

Firstly, we have to calculate the slope (m) of the line, which, for any two points \((x_1, y_1)\) and \((x_2, y_2)\), is given by the formula: \(m = \frac{{y_2 - y_1}}{{x_2 - x_1}}\). Plug the given points into the formula: \(m = \frac{{5 - (-2)}}{{-9 - 4}}\).

Step 2: Simplify the slope

Simplify the above obtained expression to find the exact value of 'm'. In doing so, we see that: \(m = \frac{{5+2}}{{-9 - 4}} = \frac{7}{-13} = -\frac{7}{13}\).

Step 3: Use the point-slope formula

Knowing that the point-slope formula is \(y - y_1 = m(x - x_1)\) (where 'm' is the slope, and \(x_1, y_1\) are the coordinates of any point on the line), we replug in any of the two points (let's say \((4, -2)\)) and 'm' into this formula: \(y - (-2) = -\frac{7}{13}(x - 4)\).

Step 4: Simplify the equation

Finally, we'll simplify the obtained equation to its simplest form: \(y + 2 = -\frac{7}{13}(x - 4)\).

Key Concepts

Slope of a Line

Understanding the slope of a line is crucial for navigating through the realm of coordinate geometry. It is essentially a measure of the steepness or the incline of a line. In algebra, the slope is often represented by the letter 'm' and it can be calculated when we have two points on a line. The formula for calculating the slope is \( m = \frac{{y_2 - y_1}}{{x_2 - x_1}} \), where \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of these points.

For example, if we have two points \( (4,-2) \) and \( (-9,5) \), we subtract the y-values and the x-values respectively to find the slope. This quantifies as a vertical change (rise) over horizontal change (run). A positive slope means the line is ascending from left to right, while a negative slope indicates a descending line. Our example yields a negative slope, indicating a downward trend as we move along the x-axis.

Linear Equations

Linear equations form the basis of a vast number of algebraic challenges. They describe a straight line on a graph and can be represented in various forms, including point-slope form, slope-intercept form, and standard form. A linear equation in two variables, x and y, has the general formula \( ax + by = c \), where 'a' and 'b' are not both zero.

The point-slope form, particularly relevant to our exercise, is expressed as \( y - y_1 = m(x - x_1) \). It directly utilises the slope of the line, along with the coordinates of a specific point on the line (\( x_1, y_1 \)). This form is incredibly useful for quickly writing equations of lines when these two pieces of information are known. Given the slope and a pair of coordinates, as shown in our example, we can swiftly derive the specific linear equation representing our line.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using the coordinate plane. This mathematical plane is defined by a horizontal axis (x-axis) and a vertical axis (y-axis), where each point has a unique set of coordinates \( (x, y) \).

Importance of Points and Lines

Central to coordinate geometry are the concepts of points and lines. Points are represented by their coordinates, while lines can be described by equations. The point-slope form of the equation of a line is an essential tool in this field, effectively linking algebra and geometry. With the coordinate geometry approach, we can solve geometric problems algebraically, such as finding the distance between two points, midpoint of a segment, and the equation of a line passing through given points.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can consist of numbers, operators, and variables. Unlike equations, expressions don't have an equal sign. They are the building blocks for forming algebraic equations, and their simplification is a fundamental aspect of algebra.

Simplification of Expressions

For instance, when we find the slope \( m = \frac{{7}}{{-13}} \) in our exercise, we simplify the expression to get a neatly defined slope. Algebraic expressions become incredibly useful when articulating relationships in formulas and solving for unknown values. Through manipulation and simplification of these expressions, we can uncover much about the relationships between the variables involved in our linear equations and ultimately solve for specific values that satisfy those equations.