Problem 26 - Algebra for College Students

Question

Find the slope of the line that contains each of the following pairs of points. \((0,0),(-2,-1)\)

Step-by-Step Solution

Answer

The slope is \(\frac{1}{2}\).

Step 1 - Understand the Slope Formula

The slope of a line that passes through two points \((x_1, y_1)\) and \((x_2, y_2)\) can be found by using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Step 2 - Identify the Coordinates

Identify the coordinates provided: \((0,0)\) and \((-2,-1)\). Let \(x_1, y_1\) be \(0,0\) and \(x_2, y_2\) be \(-2,-1\).

Step 3 - Substitute the Coordinates into the Formula

Substitute the given coordinates into the slope formula: \[ m = \frac{-1 - 0}{-2 - 0} \]

Step 4 - Simplify the Expression

Simplify the fraction: \[ m = \frac{-1}{-2} = \frac{1}{2} \]

Key Concepts

slope formula

One of the most important concepts in algebra is the slope of a line. The slope is essentially a measure of how steep a line is. To find the slope between two points on a line, you'll use the slope formula. The formula is represented as follows:

\( m = \frac{y_2 - y_1}{x_2 - x_1} \)

This formula tells you the ratio of the vertical change (rise) to the horizontal change (run) between the two points.

• \(y_2 - y_1\) represents the change in the vertical direction.
• \(x_2 - x_1\) represents the change in the horizontal direction.

So, if you have two points \((x_1, y_1)\) and \((x_2, y_2)\), plug the coordinates into the slope formula to find the slope, denoted by \(m\). Understanding this formula allows you to calculate the slope of any line, provided you have the coordinates of two points on that line.

coordinates

Coordinates are pairs of numbers that show the exact position of a point on a graph. Each coordinate is in the form of \((x, y)\), where \(x\) represents the position along the horizontal axis, and \(y\) represents the position along the vertical axis.

When solving problems involving slope, identifying the coordinates correctly is crucial. Here's an example:

• \((0,0)\): Here, both \(x\) and \(y\) are zero. This point is known as the origin.
• \((-2,-1)\): A point where \(x = -2\) and \(y = -1\).

Properly identifying coordinates ensures you do not make mistakes when applying the slope formula. By substituting the coordinates correctly, you can proceed with the mathematical operations smoothly.

simplifying fractions

Simplifying fractions is a vital step when calculating the slope of a line. In our example, we substitute the coordinates \((0,0)\) and \((-2,-1)\) into the slope formula and get:

\( m = \frac{-1 - 0}{-2 - 0} \)

To simplify this, you need to subtract and then reduce the fraction:

\( m = \frac{-1}{-2} \)

When you have a negative divided by a negative, they cancel each other out, leaving you with a positive fraction:

\( m = \frac{1}{2} \)

Remember, simplifying fractions involves:

• Performing any necessary arithmetic operations in the numerator and the denominator.
• Reducing the fraction if possible.

In our example, we only needed to recognize that a negative divided by a negative results in a positive. Simplifying fractions makes the final result easier to understand and more accurate in mathematical terms.