Problem 29 - Algebra 2

Question

Write in point-slope form the equation of the line through each pair of points. \((0,-1)\) and \((3,-5)\)

Step-by-Step Solution

Answer

The point-slope form of the equation is \( y + 1 = \frac{-4}{3}x \).

Step 1: Find the Slope (m)

Calculate the slope (m) of the line using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \), where \( (x_1, y_1) \) and \( (x_2, y_2) \) are the given points. Plug in the given points \( (0,-1) \) and \( (3,-5) \) to get \( m = \frac{-5 - (-1)}{3 - 0} = \frac{-4}{3} \).

Step 2: Use the Point-Slope Formula

The point-slope form of a line's equation is \( y - y_1 = m(x - x_1) \). Use one of the given points and the slope found in Step 1. For the point \( (0,-1) \), the equation becomes \( y - (-1) = \frac{-4}{3}(x - 0) \).

Step 3: Simplify the Equation

Simplify the equation from Step 2 to get its final point-slope form. The simplified equation is \( y + 1 = \frac{-4}{3}x \).

Key Concepts

Slope Calculation

Understanding how to calculate the slope is essential in forming the equation of a line. The slope represents how steep a line is and the direction in which the line moves, either upwards or downwards. When given two points, say \( (x_1, y_1) \) and \( (x_2, y_2) \), the slope \(m\) can be calculated using the formula \[ m = \frac{y_2 - y_1}{x_2 - x_1} \].

It involves finding the vertical change (rise) and horizontal change (run) between two points on the line. A positive slope means the line rises as it moves from left to right, while a negative slope indicates that the line falls. For the pair of points \( (0,-1) \) and \( (3,-5) \), the calculation yields a slope of \[ m = \frac{-5 - (-1)}{3 - 0} = \frac{-4}{3} \], a negative value implying a downward slope from left to right.

Point-Slope Formula

Once the slope \(m\) of a line is known, the point-slope formula can be utilized to write the equation of the line. The point-slope form is given by \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) is a specific point on the line, and \(m\) is the slope. It's a straightforward method to use when you have one point and the slope of a line.

For example, using the point \( (0,-1) \) and the slope \(\frac{-4}{3}\), we substitute into the formula to get \( y - (-1) = \frac{-4}{3}(x - 0) \), which simplifies to \( y + 1 = \frac{-4}{3}x \). This form is handy when needing to find the equation of a line quickly and is a key step in many algebra problems.

Linear Equations

Linear equations are foundational to algebra and represent lines in a coordinate system. They can take various forms, including slope-intercept \(y = mx + b\), standard \(Ax + By = C\), and point-slope \(y - y_1 = m(x - x_1)\) which we've explored with our example.

The point-slope form is especially beneficial when specific conditions like a point and slope are known. This particular form showcases the relationship of any given point on the line to the unique characteristics of the line itself — its slope. The line equation we formed earlier, \( y + 1 = \frac{-4}{3}x \) is easily convertible to other forms if needed for further analysis or graphing. Linear equations describe straight lines and are characterized by constant rates of change, which makes them predictable and reliable for calculations in various applications.