Problem 16 - Algebra 2

Question

Find the slope of the line that passes through each pair of points. $$ (-2,-3),(0,-5) $$

Step-by-Step Solution

Answer

The slope of the line is -1.

Step 1: Understand the Slope Formula

The slope (m) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1}. \] In this exercise, our points are \((-2, -3)\) and \((0, -5)\). Set \((x_1, y_1) = (-2, -3)\) and \((x_2, y_2) = (0, -5)\).

Step 2: Plug Values into the Formula

Substitute \((x_1, y_1) = (-2, -3)\) and \((x_2, y_2) = (0, -5)\) into the slope formula: \[ m = \frac{-5 - (-3)}{0 - (-2)}. \] This sets up the correct equation to find the slope using these specific points.

Step 3: Simplify the Numerator

Calculate the numerator by subtracting the y-coordinates: \(-5 - (-3)\). Simplifying gives: \(-5 + 3 = -2\).

Step 4: Simplify the Denominator

Calculate the denominator by subtracting the x-coordinates: \(0 - (-2)\). Simplifying gives: \(0 + 2 = 2\).

Step 5: Calculate the Slope

Now, with the simplified numerator and denominator, compute the slope: \[ m = \frac{-2}{2}. \] Simplifying this fraction gives \(-1\).

Step 6: Interpret the Slope

The calculated slope \(-1\) indicates that for every increase of 1 unit in the x-direction, the y-value decreases by 1 unit, showing a downward slant from left to right.

Key Concepts

Slope formula

Understanding the slope of a line is fundamental in coordinate geometry. The slope formula, \( m = \frac{y_2 - y_1}{x_2 - x_1} \), is a straightforward method to find how steep a line is between two points. The formula reveals the rate of change along a line, showing how much the y-value changes for each change in the x-value. This formula is applicable for any two points \((x_1, y_1)\) and \((x_2, y_2)\) on a line, making it a versatile tool in analyzing linear equations.

• The numerator \(y_2 - y_1\) measures how much the y-value changes.
• The denominator \(x_2 - x_1\) measures how much the x-value changes.

By dividing these changes, the slope formula gives a simple number indicating the line's steepness.

Coordinate geometry

Coordinate geometry, also known as analytic geometry, involves plotting points, lines, and shapes on a coordinate plane. By using x and y coordinates, we can analytically study geometric figures and measure distances and areas.
It allows us to apply algebraic rules to geometric problems.

• Points on a 2D plane are defined by their x (horizontal) and y (vertical) values.
• Lines and curves are represented by equations, such as linear equations.

In this exercise, we work with points \((-2, -3)\) and \((0, -5)\) on a coordinate plane to understand how position points define the line segment connecting them. Coordinate geometry forms the foundation for interpreting the slope and other geometric properties.

Linear equations

Linear equations are equations of the first degree, meaning their graph is a straight line. The standard form is \( y = mx + b \), where \(m\) represents the slope of the line, and \(b\) is the y-intercept, the point where the line crosses the y-axis.
In this context, knowing the slope \(-1\) means if we were to write an equation for the line passing through \((-2, -3)\) and \((0, -5)\), it would take the form of \( y = -x + b \). Calculating the exact value of \(b\) would require substituting one of the given points into the equation to solve for \(b\).
Linear equations are essential for describing constant rates of change and are widely used in various fields such as physics, economics, and engineering.

Slope interpretation

Understanding what a slope signifies is key to analyzing data or graphs. The slope \(m = -1\) indicates a negative relationship between the x and y axes.
Specifically, for each unit increase in x, y decreases by one unit. This negative slope results in a line that slopes downward as we move from left to right.

• Slope of \(-1\): Steepness suggests a 1:1 negative ratio between x and y changes.
• Real-world context: May represent scenarios like speed decrease or cost reductions.

Interpreting the slope helps us draw conclusions on trends and make predictions, providing valuable insights in mathematics and its applications to real-world situations.