Problem 16 - Algebra Form and Function

Question

Write an equation in point-slope form for the line. Through (4,7) and (1,1)

Step-by-Step Solution

Answer

Answer: y - 7 = 2(x - 4)

Step 1: Find the slope of the line

Using the slope formula (y2 - y1) / (x2 - x1), we will find the slope (m) of the line: m = (1 - 7) / (1 - 4) = -6 / -3 = 2

Step 2: Write the equation in point-slope form using one of the points

We can use either point (4,7) or point (1,1) to write the equation. Let's use point (4,7). The point-slope form is y - y1 = m(x - x1). Plug in the point (4, 7) and the slope 2: y - 7 = 2(x - 4) So, the equation of the line in point-slope form is: y - 7 = 2(x - 4)

Key Concepts

Equation of a Line

An equation of a line represents a straight line on a coordinate plane, describing all the points that line will pass through. There are multiple ways to write the equation of a line, but one of the most commonly used forms is the point-slope form. This is particularly useful when you know a point on the line and the slope. For instance, if a line passes through the point \((x_1, y_1)\) and has a slope \(m\), its equation in point-slope form would be \(y - y_1 = m(x - x_1)\). This format is excellent because it provides direct information about the slope and a specific point the line goes through. Other forms include the slope-intercept form \(y = mx + b\) and the standard form \(Ax + By = C\). Each format serves different applications, but the point-slope form shines in helping visualize the line's trajectory from any point given its rise and run.

Slope Formula

The slope formula is a simple yet powerful tool used in determining the steepness or incline of a line. It is calculated using the coordinates of two points on the line. The formula itself is: \[ m = \frac{y_2 - y_1}{x_2 - x_1}\]where \((x_1, y_1)\) and \((x_2, y_2)\) are two different points on the line, and \(m\) represents the slope. The slope tells us how much \(y\) changes for every unit change in \(x\). To break it down:

• \(y_2 - y_1\) is the difference in the \(y\)-coordinates, often described as the rise.
• \(x_2 - x_1\) is the difference in the \(x\)-coordinates, known as the run.

So, the slope is essentially the ratio of the vertical change to the horizontal change between two points on a line, providing insight into how steep a line is.

Linear Equations

Linear equations are equations of the first order, involving only the first powers of the variables. Such equations represent straight lines when graphed on a coordinate plane. They typically look something like \(y = mx + b\), where \(m\) is the slope of the line, and \(b\) is the y-intercept — the point where the line crosses the y-axis.What's essential to understand about linear equations is that they model a constant rate of change. Unlike quadratic or higher-degree equations, the relationship between the variables does not curve, but instead remains constant.Linear equations appear in various real-world contexts. For example:

• Calculating and predicting costs that grow at a consistent rate.
• Describing speed as a function of distance over time.

They are foundational in algebra and crucial for understanding more complex mathematical concepts. Understanding linear equations allows us to interpret and predict patterns within data, laying the groundwork for more advanced calculations.