Problem 74 - Elementary Algebra

Question

Find the slope of the line that passes through \((-8,5)\) and \((-1,-3)\)

Step-by-Step Solution

Answer

The slope of the line is \( -\frac{8}{7} \).

Step 1: Identify the coordinates

The given points are \( (-8, 5) \) and \( (-1, -3) \). Label these points as \( (x_1, y_1) \) and \( (x_2, y_2) \) respectively. So, \( x_1 = -8 \), \( y_1 = 5 \), \( x_2 = -1 \), and \( y_2 = -3 \).

Step 2: Use the slope formula

The formula to find the slope \( m \) of the line passing through 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 \( x_1 = -8 \), \( y_1 = 5 \), \( x_2 = -1 \), and \( y_2 = -3 \) into the formula:\[ m = \frac{-3 - 5}{-1 - (-8)} \]

Step 4: Simplify the numerator

Calculate \( y_2 - y_1 \):\[ -3 - 5 = -8 \]

Step 5: Simplify the denominator

Calculate \( x_2 - x_1 \):\[ -1 - (-8) = -1 + 8 = 7 \]

Step 6: Calculate the slope

Substitute the simplified numerator and denominator back into the slope formula:\[ m = \frac{-8}{7} \] Hence, the slope of the line is \( -\frac{8}{7} \).

Key Concepts

Understanding Coordinates

Coordinates are a pair of numbers that locate a point on a graph. They are often written as \((x, y)\). The first number, $$x$$, is the horizontal position. The second number, $$y$$, is the vertical position. In this problem, the coordinates given are \((-8, 5)\) and \((-1, -3)\).

To solve problems related to slopes, label the coordinates correctly. Usually, we use \((x_1, y_1)\) for the first point and \((x_2, y_2)\) for the second point. This helps avoid confusion when plugging numbers into formulas. Remember, getting the coordinates right is crucial for accurate calculations!

Using the Slope Formula

The slope of a line shows its steepness. To find the slope, we use the slope formula:

$$\[m = \frac{y_2 - y_1}{x_2 - x_1}\]$$
The slope formula requires two coordinates, \((x_1, y_1)\) and \((x_2, y_2)\). It's essential to subtract the respective $$y$$-values and $$x$$-values correctly.

In our example, substitute the values into the formula:
\( y_2 - y_1 = -3 - 5 = -8 \)
\( x_2 - x_1 = -1 - (-8) = -1 + 8 = 7 \)

We then divide the two results to find the slope. Thus the slope $$m$$ is $$\[m = \frac{-8}{7}\]$$. Practicing the use of the slope formula with different coordinates helps master this method.

Simplifying Fractions

Often in math, you'll need to simplify fractions to their lowest terms. This makes them easier to understand and use. Let's simplify our slope:

The fraction $$\frac{-8}{7}$$ cannot be simplified further because 8 and 7 have no common factors other than 1.

Here are tips for simplifying fractions:

• Identify common factors of the numerator and denominator.
• Divide the numerator and denominator by their greatest common divisor (GCD).

If your fraction can't be simplified, like in this case, it’s already in its simplest form. Practicing simplification helps make calculations easier and faster.