Question

Find the slope of the line joining the points (2,5) and (-1,3).

Short answer

The slope is \(\frac{2}{3}\).

Step-by-step solution

Identify the coordinates

First, note the coordinates of the given points. The points are (2,5) and (-1,3). Let (x1, y1) be (2, 5) and (x2, y2) be (-1, 3).

Understand the slope formula

Recall that the slope of a line passing through two points \( (x1, y1) \) and \( (x2, y2) \) is calculated using the formula \(\frac{y2 - y1}{x2 - x1}\).

Substitute the coordinates into the slope formula

Substitute \( x1 = 2 \), \( y1 = 5 \), \( x2 = -1 \), and \( y2 = 3 \) into the slope formula: \(\frac{3 - 5}{-1 - 2}\).

Simplify the expression

Simplify the numerator and the denominator: \(\frac{3 - 5}{-1 - 2} = \frac{-2}{-3}\).

Reduce the fraction

Finally, reduce the fraction \(\frac{-2}{-3}\) to get \(\frac{2}{3}\).

Key concepts

coordinates

Coordinates are a set of values that show an exact position. In a two-dimensional space, these are often written as \( (x, y) \). The first number, \( x \), represents the horizontal position, and the second number, \( y \), represents the vertical position.
In our exercise, the coordinates of the points are (2,5) and (-1,3).
Understanding coordinates is key to solving many geometry problems.
Here are some more tips to remember about coordinates:

• The horizontal axis (x-axis) moves left and right.
• The vertical axis (y-axis) moves up and down.
• Positive x-values are to the right, and negative x-values are to the left.
• Positive y-values are upwards, and negative y-values are downwards.

By knowing the coordinates of points, we can easily find their positions in a plane. This helps us significantly when calculating distances or slopes.

slope formula

\( \text{slope} = \frac{3 - 5}{-1 - 2} \)
Next, we solve the numerator and the denominator separately:

• Numerator: \( 3 - 5 = -2 \).
• Denominator: \( -1 - 2 = -3 \).

So, the slope is \[ \frac{-2}{-3} \].

simplifying fractions

Simplifying fractions is a process of making the fraction as simple as possible. When it comes to slopes, this usually means reducing the fraction to its simplest form.
Following our previous steps, we have the fraction \( \frac{-2}{-3} \).
The negative signs in the numerator and denominator cancel each other out because a negative divided by a negative is a positive. Thus, we can simplify like this:

• \( \frac{-2}{-3} = \frac{2}{3} \).

Now, our slope, \( \frac{2}{3} \), is in its simplest form.
Simplifying fractions for slope is important to make the value more understandable and usable in further calculations.
Remember the general steps for simplifying fractions:

• Find the greatest common factor (GCF) of the numerator and the denominator.
• Divide both the numerator and the denominator by the GCF.
• If both the numerator and the denominator are negative, cancel out the negative signs.

By following these steps every time, you can ensure your fractions are always in their simplest form.