Problem 6 - Elementary Algebra

Question

Find the slope of the line determined by each pair of points. $$(3,9),(8,4)$$

Step-by-Step Solution

Answer

The slope is -1.

Step 1: Identifying Points

We are given two points on a line: \((x_1, y_1) = (3, 9)\) and \((x_2, y_2) = (8, 4)\).

Step 2: Understanding the Slope Formula

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

Step 3: Substituting Values

Substitute the coordinates of the given points into the slope formula. This gives us: \[ m = \frac{4 - 9}{8 - 3} \]

Step 4: Simplifying the Expression

Calculate the difference in the y-values and in the x-values separately: \[ 4 - 9 = -5 \] and \[ 8 - 3 = 5 \] Then substitute these results into the formula: \[ m = \frac{-5}{5} \]

Step 5: Calculating the Slope

Simplify the fraction: \[ m = -1 \] Therefore, the slope of the line is \( -1 \).

Key Concepts

Coordinates

Coordinates are an essential aspect of understanding geometry and algebra. They represent the position of a point on a plane, described by an ordered pair \((x, y)\). The first number in the pair is the x-coordinate, which is the horizontal position of the point. The second number is the y-coordinate, describing the vertical position. In this exercise, we have two points: \((3, 9)\) and \((8, 4)\). These coordinates tell us exactly where each point lies on a coordinate plane. If you imagine a giant grid, the x-coordinate indicates how far across the grid you go, and the y-coordinate shows how far up or down.

• For \((3, 9),\) you move 3 units right and 9 units up.
• For \((8, 4),\) you move 8 units right and 4 units up.

Coordinates are the basis for finding other important characteristics of lines, such as their slope.

Slope Formula

The slope formula is a mathematical tool used to determine how steep a line is between two designated points. It's essentially a measure of how much a line inclines or declines as it moves across the coordinate plane. The slope formula is expressed as \( m = \frac{y_2 - y_1}{x_2 - x_1} \).

• \( m \) represents the slope.
• \( y_2 \) and \( y_1 \) are the y-coordinates of the two points.
• \( x_2 \) and \( x_1 \) are the x-coordinates of the two points.

In practice, you subtract the y-coordinate of the first point from the y-coordinate of the second point. Then, do the same with the x-coordinates.
For the points \((3, 9)\) and \((8, 4)\), we plug into the slope formula:\[m = \frac{4 - 9}{8 - 3}\]This formula helps you see the rate of change, or how fast one variable changes with respect to another. A positive slope means the line rises as it moves from left to right, while a negative slope indicates it falls.

Line Equation

The equation of a line is a fundamental concept connecting geometry with algebra.Once we know the slope and a point on the line, we can express the line in its equation form, usually the slope-intercept form: \( y = mx + b \). Here,

• \( m \) is the slope of the line.
• \( b \) is the y-intercept, which is where the line crosses the y-axis.

To find \( b \), you can plug one of the points and the slope into the equation and solve for \( b \).
For our example, with a slope \( m = -1 \) and using the point \((3, 9)\), you'll substitute into the line equation:\( 9 = -1(3) + b \).Solving for \( b \) gives us:\[9 = -3 + b \]Adding 3 to both sides,\[12 = b\]The final equation of the line would be:\[ y = -1x + 12 \]This shows that the line decreases 1 unit vertically for each unit it moves horizontally, starting at 12 when \( x \) is 0. This way, we can easily graph or understand the behavior of the line on a plane.