Problem 57 - Algebra A Combined Function

Question

Write an equation of each line. Write the equation in standard form unless indicated otherwise. See Examples 1 through \(6 .\) Through (2,-8) and (-6,-5)\(;\) use slope-intercept form.

Step-by-Step Solution

Answer

The equation is \(y = -\frac{3}{8}x - \frac{29}{4}\).

Step 1: Identify the given points

We are given two points: \((2, -8)\) and \((-6, -5)\). These points will help us find the slope of the line and then the equation in slope-intercept form.

Step 2: Calculate the slope of the line

The slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Substituting the given points into the formula gives us:\[m = \frac{-5 - (-8)}{-6 - 2} = \frac{-5 + 8}{-6 - 2} = \frac{3}{-8} = -\frac{3}{8}\]

Step 3: Use the slope-point form to find the equation

The slope-point form of a line's equation is \(y - y_1 = m(x - x_1)\). Using the slope \(-\frac{3}{8}\) and point \((2, -8)\), substitute them into the formula:\[y - (-8) = -\frac{3}{8}(x - 2)\]which simplifies to:\[y + 8 = -\frac{3}{8}(x - 2)\]

Step 4: Expand and simplify to get the slope-intercept form

Now distribute \(-\frac{3}{8}\) in the equation:\[y + 8 = -\frac{3}{8}x + \frac{3 \times 2}{8}\]Simplify it:\[y + 8 = -\frac{3}{8}x + \frac{6}{8}\]\[y + 8 = -\frac{3}{8}x + \frac{3}{4}\]Subtract 8 from both sides to isolate \(y\):\[y = -\frac{3}{8}x + \frac{3}{4} - 8\]Convert \(-8\) to a fraction with denominator 4:\[y = -\frac{3}{8}x + \frac{3}{4} - \frac{32}{4}\]Combine the fractions:\[y = -\frac{3}{8}x - \frac{29}{4}\]

Step 5: Final Step: Write the equation in slope-intercept form

The final equation of the line in slope-intercept form is:\[y = -\frac{3}{8}x - \frac{29}{4}\]

Key Concepts

Equation of a Line

The equation of a line is a mathematical statement that describes a straight line on a coordinate plane. It tells us the relationship between the x and y coordinates of any point on the line. There are different forms to express this equation, such as the standard form and the slope-intercept form. In this context, we focus on the slope-intercept form, represented as \(y = mx + b\), where:- \(m\) is the slope of the line.- \(b\) is the y-intercept, which is the point where the line crosses the y-axis.This form is particularly useful because it gives a direct understanding of how steep the line is and where it starts on the y-axis.
In problems like the one given, you're often asked to find the equation of a line using specific points. Let these points be \((x_1, y_1)\) and \((x_2, y_2)\). To derive the equation, you typically need to determine the slope first and then use it along with one point to find the equation in slope-intercept form.

Slope Calculation

The slope of a line is a number that describes how slanted the line is. It tells us the change in the y-direction (vertical) per unit change in the x-direction (horizontal). This is typically written as \(m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}\).
In this particular exercise, we're given two points: \((2, -8)\) and \((-6, -5)\). To find the slope between these two points, we substitute them into the slope formula:- \(y_2 - y_1 = -5 - (-8) = -5 + 8 = 3\)- \(x_2 - x_1 = -6 - 2 = -8\)Thus, the slope \(m\) is calculated as \(\frac{3}{-8} = -\frac{3}{8}\).
Remember, the slope is negative, indicating that as you move from left to right, the line goes downwards. Always make sure to carefully compute the differences in y and x, as reversing these can lead to incorrect answers.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This approach allows us to describe and analyze geometric figures algebraically using concepts like lines, points, and slopes in a two-dimensional plane. For instance:- You have coordinates \((x, y)\) that locate points on the plane.- Equations define lines or curves in these terms, offering a way to manipulate and understand geometric figures mathematically.
In the exercise where you find the equation of a line through two points, coordinate geometry plays a central role. The points \((2, -8)\) and \((-6, -5)\) are specified in terms of their coordinates. The beauty of coordinate geometry is that it bridges algebra and geometry, allowing us to derive the equation of a line from just two of its points.
Understanding coordinate geometry equips you with the tools to solve complex problems involving shapes and distances analytically rather than just visually.