Problem 64 - Algebra 2

Question

Find the slope of the line that passes through each pair of points. $$ (1,0),(-2,9) $$

Step-by-Step Solution

Answer

The slope is \(-3\).

Step 1: Identify the Given Points

We are given two points, \((x_1, y_1) = (1, 0)\) and \((x_2, y_2) = (-2, 9)\).

Step 2: Recall the Formula for Slope

The formula to calculate the slope \(m\) of a line through two points \((x_1, y_1)\) and \((x_2, y_2)\) is \( m = \frac{y_2 - y_1}{x_2 - x_1} \).

Step 3: Substitute the Values into the Slope Formula

Substitute the given values into the formula: \( m = \frac{9 - 0}{-2 - 1} \).

Step 4: Calculate the Differences

Calculate the difference in the numerator: \(9 - 0 = 9\). Calculate the difference in the denominator: \(-2 - 1 = -3\).

Step 5: Perform the Division

Divide the results from the numerator by the denominator: \( m = \frac{9}{-3} \).

Step 6: Simplify the Slope

Simplify the fraction \( \frac{9}{-3} \) to get \( -3 \).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a powerful mathematical tool that allows us to describe geometric figures algebraically. By using a coordinate plane with an x-axis and a y-axis, we can plot points using ordered pairs \((x, y)\).

This approach makes it easier to calculate distances, slopes, and other geometrical properties. For example, in the given exercise, two points \((1,0)\) and \((-2,9)\) were plotted on this coordinate grid. Understanding how these points relate to each other on the plane is essential when solving for the slope of the line they form.

Coordinate geometry provides many advantages:

• Helps visualize mathematical concepts such as lines and curves.
• Allows the use of algebraic methods to solve geometric problems.
• Makes it simple to identify intercepts and other key points on a line or curve.

By using coordinate geometry, we bridge the gap between algebra and geometry, making complex problems easier to solve and understand.

Point-Slope Formula

The point-slope formula is a common tool for writing the equation of a line when you know the slope and at least one point on the line. Given a point \((x_1, y_1)\) and a slope \(m\), the equation of a line can be expressed as:

\[y - y_1 = m(x - x_1)\]

This formula is particularly useful when dealing with real-world problems where measuring slope directly may be challenging. Let's apply the point-slope formula in the context of the original exercise to better understand it.

Suppose we found the slope \(m = -3\) and we have a point \((1,0)\). We can plug these into the formula as follows:

• Substitute \(x_1 = 1\), \(y_1 = 0\), and \(m = -3\) into the formula.
• The equation becomes: \[y - 0 = -3(x - 1)\]
• Simplify the equation: \[y = -3x + 3\]

Now, the line equation in point-slope form helps us understand the relationship between the x and y variables, making it straightforward to calculate any point along the line.

Linear Equations

Linear equations are algebraic expressions that describe a straight line through their graphical representation. In the standard form, a linear equation is typically written as:

\[Ax + By = C\]

However, for simplicity and direct application of slope, we often use the slope-intercept form:

\[y = mx + b\]

Here, \(m\) is the slope, and \(b\) is the y-intercept, the point where the line crosses the y-axis.

In the context of the given exercise, after finding the slope as \(-3\), we can convert the point-slope form of the equation to this slope-intercept form:

• We previously derived the equation: \[y = -3x + 3\].
• Here, \(-3\) represents the slope \(m\), and \(3\) represents the y-intercept \(b\).

Linear equations are fundamental because they model relationships with constant rates of change. They are everywhere in fields like physics, engineering, economics, and more, helping to describe how one variable changes with respect to another.