Problem 32 - Beginning and Intermediate Algebra

Question

Use the slope formula to find the slope of the line containing each pair of points. $$(-7,4) \text { and }(1,4)$$

Step-by-Step Solution

Answer

The slope of the line containing the points \((-7,4)\) and \((1,4)\) is 0.

Step 1: Identify the coordinates of the points

We are given the points \((-7,4)\) and \((1,4)\). Identify the coordinates as follows: Point 1: \((x1, y1) = (-7, 4)\) Point 2: \((x2, y2) = (1, 4)\)

Step 2: Apply the slope formula

Now, apply the slope formula using the identified coordinates. So, we have: \(m = \frac{y2-y1}{x2-x1}\)

Step 3: Substitute the coordinates into the formula

Substitute the coordinates from step 1 into the slope formula: \(m = \frac{4-4}{1-(-7)}\)

Step 4: Simplify the expression

Simplify the expression to find the slope: \(m = \frac{0}{8}\)

Step 5: Calculate the slope

Now, we can compute the slope by dividing the numerator by the denominator: \(m = 0\) The slope of the line containing the points \((-7,4)\) and \((1,4)\) is 0.

Key Concepts

Coordinate Geometry

In coordinate geometry, we look at shapes and lines in a grid-like setup known as the Cartesian plane. This plane is made from two perpendicular lines named the x-axis and y-axis. Here's where we plot points that have coordinates such as \((-7,4)\) and \(1,4)\).
The coordinates are like directions on the grid:

• The first number of the pair tells us how far along the x-axis (the horizontal line) we move.
• The second number tells us how far up or down the y-axis (the vertical line) we go.

Points could form lines or curves and are essential for drawing graphs of equations. In this exercise, we're given two points which will help us determine the nature of the line when connected. Understanding coordinate geometry makes it easier to visualize and solve problems involving these points.

Slope Formula

The slope formula gives us a way to calculate the steepness or tilt of a line. It's like measuring the angle of a hill to see how steep it is. The formula is as follows:

• \( m = \frac{\Delta y}{\Delta x} \) which means the change in y-coordinates over the change in x-coordinates.

Using the points from the exercise, let's see how it works. We were given two points \((-7,4)\) and \(1,4)\), and the slope formula becomes:

• Find the difference in y-coordinates: \(4-4 = 0\)
• Find the difference in x-coordinates: \(1 - (-7) = 8\)

When we plug these into our formula, it becomes \( m = \frac{0}{8} \).
The slope here is \(0\). A zero slope tells us the line is absolutely flat, just like a table. This means no matter where you look along the x-axis, the y-value doesn't change—perfectly horizontal!

Linear Equations

Linear equations represent straight lines when graphed on the coordinate plane. They're often written in the format \(y = mx + b\), where:

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

In our example, we discovered the slope is \(0\). This means the equation looks something like \(y = 0x + b\) or simply \(y = b\).
If both points \((-7,4)\) and \(1,4)\) have a y-coordinate of \(4\), then \(b\), the y-value where the line crosses the y-axis is \(4\). So, the equation of our line is \(y = 4\).
Linear equations help us predict and understand relationships. In this case, it tells us that no matter how far we travel along the x-axis, the y-value remains steady at 4, ensuring a constant level. Linear equations are powerful tools in algebra and beyond.