Problem 67 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The slope is \(\frac{5}{2}\).

Step 1: Identify the Two Points

The first point is \((0, 3)\) and the second point is \((-2, -2)\).

Step 2: Recall the Slope Formula

The formula to find the slope \(m\) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Step 3: Substitute the Coordinates into the Formula

Using the points \((0, 3)\) and \((-2, -2)\), substitute into the slope formula:\[ m = \frac{-2 - 3}{-2 - 0} \]

Step 4: Simplify the Numerator

Calculate the difference in the \(y\)-coordinates: \[ -2 - 3 = -5 \]

Step 5: Simplify the Denominator

Calculate the difference in the \(x\)-coordinates: \[ -2 - 0 = -2 \]

Step 6: Calculate the Slope

Substitute the simplified values back into the formula:\[ m = \frac{-5}{-2} = \frac{5}{2} \]

Key Concepts

Coordinate Geometry

Coordinate geometry is a branch of mathematics that allows you to explore geometric figures using a coordinate system. It's like plotting maps, but instead of streets and buildings, you plot shapes and figures. This method uses numbers (coordinates) to represent points on a plane, making it easier to analyze geometric shapes. The basic components are the x-axis and y-axis, which intersect at the origin (0,0). Every point in this system is represented by a pair of numbers \(x, y\). Using this system, we can determine relationships such as distance, midpoint, and the slope of a line, which measures how steep the line is. By working with these components, coordinate geometry helps us solve practical problems easily.

Slope Formula

The slope formula is a crucial tool in coordinate geometry. It helps in determining the steepness or incline of a line. This formula is valuable when you have two points and need to find how the line connects them.

• The slope (m) is calculated as the ratio of the vertical change to the horizontal change between two points.

For two points, say, \(x_1, y_1\) and \(x_2, y_2\), the slope formula is:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]This formula provides a precise measure:

• A positive slope suggests the line goes upwards from left to right.
• A negative slope means it goes downwards.
• If the slope is zero, the line is flat.
• A vertical line has an undefined slope since the denominator of the formula becomes zero (division by zero).

Understanding the slope formula is essential for graph reading and creating slopes in algebra.

Points on a Line

Points on a line in the coordinate plane are critical for defining the line's path. Each point has coordinates that tell exactly where it is located.

• The first number of the pair, (\(x\)), refers to the horizontal position.
• The second number, (\(y\)), represents the vertical position.

When examining a line, any two points are enough to describe its direction and steepness using the slope formula. In practical terms, these points act like pegs on a string, defining its tightness when pulled taut, and determining its direction on the plane. This concept ensures that if one knows two distinct points on a line, they can map out the entire line.

Algebraic Calculations

Algebraic calculations in geometry often involve substituting values and simplifying expressions to find unknown variables. Specifically, when calculating the slope from given points, you perform certain algebraic steps.

• First, you substitute the y-coordinates and x-coordinates of the points into the slope formula.
• Then, solve the differences (subtractions) in both numerator and denominator separately.
• Finally, divide the simplified values to obtain the slope.

In the given problem, the points \(0, 3\) and \(-2, -2\) are substituted into the formula. The calculation follows:

• Numerator: \(-2 - 3 = -5\)
• Denominator: \(-2 - 0 = -2\)
• Slope: \(\frac{-5}{-2} = \frac{5}{2}\)

Algebraic calculations help simplify these steps and solve for slopes and other measurements easily, providing clear insights into geometric interpretations.