Problem 28 - Algebra for College Students

Question

Find the slope of the line that contains each of the following pairs of points. \((3,0),(0,10)\)

Step-by-Step Solution

Answer

The slope is \( -\frac{{10}}{{3}} \).

Step 1 - Identify the coordinates

Identify the coordinates of the points given. These points are \(3,0\) and \(0,10\). Here, the coordinates are \((x_1, y_1) = (3, 0)\) and \((x_2, y_2) = (0, 10)\).

Step 2 - Recall the slope formula

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

Step 3 - Substitute the coordinates into the formula

Substitute the coordinates \(3, 0\) and \(0, 10\) into the slope formula: \[ m = \frac{{10 - 0}}{{0 - 3}} \]

Step 4 - Simplify the expression

Simplify the expression to find the slope: \[ m = \frac{{10}}{{-3}} = -\frac{{10}}{{3}} \]

Key Concepts

Slope Formula

Understanding how to find the slope of a line is crucial in coordinate geometry. The slope formula is used to determine the steepness or incline of a line. For any two points \((x_1, y_1)\) and \((x_2, y_2)\), the slope \(m\) is calculated as follows: \[ m = \frac{{y_2 - y_1}}{{x_2 - x_1}} \]. This ratio \(\frac{{\Delta y}}{{\Delta x}}\), where \(\Delta y\) represents the change in y-coordinates and \(\Delta x\) the change in x-coordinates, tells us how much the y-coordinate changes for a unit change in the x-coordinate. To apply this formula accurately, you need to correctly substitute the coordinates of the points into the equation and perform the necessary arithmetic operations.

Coordinate Geometry

In coordinate geometry, points, lines, and shapes are represented on a coordinate plane. This plane is defined by two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). Points on this plane are identified by their coordinates \((x, y)\), which represent their horizontal and vertical distances from the origin \((0, 0)\). Understanding the layout of the coordinate plane makes it easier to visualize and solve problems involving distance, midpoint, and slope. When given two points, you can use their coordinates to draw the line that connects them and then apply the slope formula to understand the direction and steepness of that line.

Linear Equations

Linear equations describe straight lines on the coordinate plane. The general form of a linear equation is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept (the point where the line crosses the y-axis). The slope \(m\) indicates the direction and steepness of the line. Understanding how to manipulate and solve linear equations is crucial for graphing and finding relationships between variables. For instance, once you know the slope from two points, you can use it to write the linear equation of the line that passes through those points by using the point-slope formula \[ y - y_1 = m(x - x_1) \]. This allows you to convert between different forms of linear equations seamlessly.