Problem 14 - Algebra 2

Question

Find the slope of the line that passes through each pair of points. $$ (-8,-3),(2,3) $$

Step-by-Step Solution

Answer

The slope is \(\frac{3}{5}\).

Step 1: Recall the Slope Formula

The slope of a line passing through two points \(x_1, y_1\) and \(x_2, y_2\) is given by the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). This formula calculates the change in \(y\) divided by the change in \(x\).

Step 2: Identify the Coordinates

Here, the two points given are \((-8, -3)\) and \(2, 3\). Thus, \(x_1 = -8\), \(y_1 = -3\), \(x_2 = 2\), and \(y_2 = 3\).

Step 3: Substitute the Values

Substitute the values into the slope formula: \(\begin{align*}m &= \frac{y_2 - y_1}{x_2 - x_1} \&= \frac{3 - (-3)}{2 - (-8)}\end{align*}\).

Step 4: Simplify the Expression

Further simplify the expression: \(\begin{align*}m &= \frac{3 + 3}{2 + 8} \&= \frac{6}{10}\end{align*}\).

Step 5: Simplify the Fraction

Divide both the numerator and the denominator by their greatest common divisor, which is 2: \(m = \frac{6}{10} = \frac{3}{5}\). Thus, the slope of the line is \(\frac{3}{5}\).

Key Concepts

Slope Formula

The slope formula is key to understanding how steep a line is. It tells us how much the line rises or falls for every unit it moves horizontally. The formula for slope, often noted as \( m \), is given by:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]Here:

• \( x_1, y_1 \) are the coordinates of the first point on the line,
• \( x_2, y_2 \) are the coordinates of the second point.

Using the formula, you can find the change in the vertical direction (\( y \)) and divide it by the change in the horizontal direction (\( x \)). This ratio gives the slope of the line.

Coordinates

Coordinates are used to pinpoint the exact location of a point on a graph. They come in pairs, written as \((x, y)\), and represent positions on the x-axis (horizontal) and y-axis (vertical), respectively.For example, the point \((-8, -3)\) means:

• The point is 8 units left of the origin on the x-axis because of the negative sign.
• It is 3 units down on the y-axis for the same reason.

These coordinate points help us determine where the line starts and ends, which is essential for calculating slope using the formula.

Simplifying Fractions

Simplifying fractions makes them easier to understand and work with. A fraction is simplified when it has the smallest possible numbers on the top (numerator) and bottom (denominator).For instance, when you have the fraction \(\frac{6}{10}\), it's not in its simplest form. By dividing both numbers by 2, you reduce it to \(\frac{3}{5}\). Simplifying shouldn't change the value of the fraction, just make it as simple as possible. This reduced form of fractions provides a clearer picture of the slope.

Greatest Common Divisor

The greatest common divisor (GCD) is the largest number that divides both the numerator and the denominator without leaving a remainder.To simplify fractions like \(\frac{6}{10}\), you need the GCD of 6 and 10, which is 2. Here's how it works:

• List the factors of 6: 1, 2, 3, 6.
• List the factors of 10: 1, 2, 5, 10.
• Identify the largest number they both have: 2.

Using the GCD, divide both the numerator and denominator to simplify the fraction, making it easier to work with in problems.