Problem 24 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems \(23-28\), find the slope of the line containing the given two points. (3,5) \text { and }(4,7)

Step-by-Step Solution

Answer

The slope of the line is 2.

Step 1: Identify the Points

The problem gives us two points: \((3, 5)\) and \((4, 7)\). We need to use these points to find the slope of the line that passes through them.

Step 2: Recall the Slope Formula

The formula to find the slope \(m\) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Step 3: Substitute the Points into the Slope Formula

Substitute \((x_1, y_1) = (3, 5)\) and \((x_2, y_2) = (4, 7)\) into the formula: \[ m = \frac{7 - 5}{4 - 3} \]

Step 4: Calculate the Differences in Coordinate Values

Calculate \(y_2 - y_1\): \(7 - 5 = 2\). Calculate \(x_2 - x_1\): \(4 - 3 = 1\).

Step 5: Compute the Slope

Substitute the differences into the slope formula to find: \[ m = \frac{2}{1} = 2 \] Thus, the slope of the line is 2.

Key Concepts

Slope Formula

The slope of a line is a measure of how steep the line is. It represents the change in the vertical direction (up and down) compared to the change in the horizontal direction (left and right). Understanding how to calculate the slope is crucial, especially when dealing with linear equations and graphs. To find the slope of a line passing through two points, you use the slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Here, \(m\) represents the slope, \(x_1, y_1\) are the coordinates of the first point, and \(x_2, y_2\) are the coordinates of the second point. What this formula essentially does is calculate the ratio of the "rise" (change in \(y\) values) over the "run" (change in \(x\) values). A positive slope means the line is rising, a zero slope means it's flat, and a negative slope means it's falling as you move from left to right along the line.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This form of geometry allows us to use algebraic techniques to solve geometric problems. In coordinate geometry, we plot points on a flat surface with two axes: a horizontal axis, often called the \(x\)-axis, and a vertical axis, known as the \(y\)-axis. By using coordinates \( (x, y) \), we can precisely locate or describe the position of any point on this plane.

• The intersection of the \(x\)-axis and \(y\)-axis is called the origin, represented by the point \( (0, 0) \).
• Knowing the coordinates of two points helps in finding distances between them, calculating slopes, and deriving equations of lines.

Coordinate geometry is a powerful tool because it bridges algebra and geometry, allowing us to use computational methods to solve problems involving shapes and spaces.

Linear Equations

Linear equations are algebraic expressions where every term is either a constant or the product of a constant and a single variable. In its simplest form, a linear equation can have one or two variables, and its graph is always a straight line. For a linear equation with two variables \(x\) and \(y\), it is often written in the slope-intercept form: \[ y = mx + b \] Here, \(m\) is the slope of the line, and \(b\) represents the \(y\)-intercept, which is the point where the line crosses the \(y\)-axis. This form is particularly useful as it directly tells us the slope of the line and allows us to easily graph it by starting at the \(y\)-intercept and using the slope to find other points. Linear equations are foundational in calculus and many areas of mathematics because they describe relationships with constant rates of change. Understanding how to interpret and graph these equations is critical for solving many real-world problems, such as predicting trends and analyzing rates.