Problem 60 - Algebra A Combined Function

Question

Find the slope of the line containing each pair of points. (-5,-2),(0,7)

Step-by-Step Solution

Answer

The slope is \( \frac{9}{5} \).

Step 1: Understanding the Slope Formula

The slope (m) of a line is determined by the formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \) where \((x_1, y_1)\) and \((x_2, y_2)\) are two distinct points on the line.

Step 2: Identifying the Points

Identify the given points from the problem. The points are \((-5, -2)\) and \((0, 7)\). We assign them as follows: \(x_1 = -5\), \(y_1 = -2\), \(x_2 = 0\), \(y_2 = 7\).

Step 3: Substituting into the Formula

Substitute the values into the slope formula: \( m = \frac{7 - (-2)}{0 - (-5)} \).

Step 4: Simplifying the Expression

Calculate the differences in the numerator and denominator: \( m = \frac{7 + 2}{0 + 5} = \frac{9}{5} \).

Step 5: Finalizing the Slope

The slope of the line is \( \frac{9}{5} \). This means that for every 5 units increase in x, y increases by 9 units.

Key Concepts

Slope of a Line

In algebra, the slope of a line is a key concept that helps in understanding how a line inclines or declines. Imagine you are skiing down a hill. The steeper the hill, the greater the slope. Likewise, on a graph, the slope indicates how much y changes for a change in x.

To determine the slope between two points on a line, the formula used is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). This formula requires two points so you can understand how the line rises or falls as it moves from left to right.

Some key points about slope include:

• If the slope is positive, the line goes upwards as it moves right.
• If the slope is negative, the line goes downwards as it moves right.
• If the slope is zero, the line is flat or horizontal.
• Infinite or undefined slope occurs in vertical lines, where x remains constant.

Coordinate Points

Coordinate points are essential in graphing and algebra because they help in identifying specific positions on a graph. Each coordinate consists of an ordered pair \((x, y)\), where \(x\) represents the horizontal position, and \(y\) refers to the vertical position in a 2-dimensional space.

For example, in the given problem, the points \((-5, -2)\) and \((0, 7)\) locate specific positions on the graph. Knowing these, you can determine the slope. These points tell not just where the line is but also how it behaves along the coordinate plane.

Here's how to extract information from coordinate points:

• The first number in the pair (\(x\)) is how far you go left or right.
• The second number (\(y\)) is how far you go up or down.
• Coordinates help you "see" the equation graphically.
• Identifying the correct \((x, y)\) pairs is crucial before using any formula.

Algebraic Formula

In algebra, formulas serve as a bridge between abstract algebraic expressions and their geometric interpretations. An algebraic formula is an equation that represents relationships between different variables.

The slope formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\) is a practical example. It helps in switching from just numbers to understanding a geometric concept on a graph. Before solving, you should identify the coordinates and substitute them into the algebraic formula. This step ensures that your calculations reflect the correct rise and run between points.

When dealing with algebraic formulas, remember:

• Substitute correctly; ensure each value goes into the right place.
• Follow order of operations to avoid mistakes.
• Simplify expressions carefully to derive meaningful results.
• Formulas often represent real-world relationships in mathematical terms.