Problem 27 - Algebra 1

Question

Write an equation in point-slope form of the line that passes through the given points. $$ (-3,-9),(-6,-8) $$

Step-by-Step Solution

Answer

The equation of the line in point-slope form that passes through the points (-3,-9) and (-6,-8) is \( y = 1/3x -1 \)

Step 1: Find the slope m

Use the formula for the slope, m = (y_2 - y_1) / (x_2 - x_1). Substitute the coordinates of the given points into the formula, which results in m = (-8 - -9) / (-6 - -3) = 1/3.

Step 2: Substitute the slope and any point into the point-slope form

Use the point (-3, -9) and the slope 1/3 to substitute into the point-slope form \(y - y_1 = m(x - x_1)\). The resulting equation is \(y - -9 = 1/3(x - -3)\). This simplifies to \(y + 9 = 1/3(x + 3)\).

Step 3: Simplify the equation

The final form of the equation after simplification is \(y = 1/3x -1 \).

Key Concepts

Slope Calculation

Understanding the concept of slope is key when dealing with linear equations. It indicates the steepness or tilt of a line, and is often denoted by the letter \( m \). The formula for calculating the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

To apply this, simply substitute the coordinates of the given points into the formula. For instance, given points \((-3, -9)\) and \((-6, -8)\), the calculation is:

• \( m = \frac{-8 - (-9)}{-6 - (-3)} = \frac{1}{3} \)

Thus, the slope \( m \) of the line passing through these points is \( \frac{1}{3} \). This slope tells us that for every increase of 1 in the \( x \)-value, the \( y \)-value increases by \( \frac{1}{3} \).

Equation Simplification

Equation simplification involves reducing an equation to its simplest form. Once you've determined the slope and chosen a point from your data, you substitute these into the point-slope form of an equation, which is:

• \( y - y_1 = m(x - x_1) \)

Take the slope \( \frac{1}{3} \) and the point \((-3, -9)\), and substitute them into the formula:

• \( y - (-9) = \frac{1}{3}(x - (-3)) \)

This can be simplified to:

• \( y + 9 = \frac{1}{3}(x + 3) \)

Further simplifying aims to isolate \( y \) on one side of the equation:

• Distribute the \( \frac{1}{3} \) over the expression \( (x + 3) \): \( y + 9 = \frac{1}{3}x + 1 \)

Subtracting 9 from both sides gives the simplified equation:

• \( y = \frac{1}{3}x - 1 \)

This is the linear equation in its simplest form.

Linear Equations

Linear equations represent straight lines and are a fundamental part of algebra. They are used to model relationships between two variables. The general form of a linear equation is:

• \( y = mx + b \)

Where \( m \) represents the slope, and \( b \) is the y-intercept, indicating where the line crosses the y-axis.
Linear equations can be expressed in different forms, such as point-slope form, slope-intercept form, or standard form. Each form serves a particular purpose:

• Point-Slope Form: Useful for creating an equation when you know one point and the slope.
• Slope-Intercept Form: Allows immediate identification of the slope and y-intercept. It's very user-friendly for graphing.
• Standard Form: Often used in integer form for linear optimization.

In this exercise, we derived a point-slope equation \( y + 9 = \frac{1}{3}(x + 3) \) and simplified it to \( y = \frac{1}{3}x - 1 \), illustrating how linear equations are versatile and crucial in mathematical analysis.