Problem 32 - College Algebra with Corequisite Support

Question

For the following exercises, given each set of information, find a linear equation satisfying the conditions, if possible. Passes through (1,5) and (4,11)

Step-by-Step Solution

Answer

The linear equation is \(y = 2x + 3\).

Step 1: Identify the Points

We are given two points through which the line passes: the points are \((1, 5)\) and \((4, 11)\).

Step 2: Calculate the Slope

The formula for the slope \(m\) is given by \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Plugging in our points \((x_1, y_1) = (1, 5)\) and \((x_2, y_2) = (4, 11)\), we have:\[ m = \frac{11 - 5}{4 - 1} = \frac{6}{3} = 2 \].The slope of the line is 2.

Step 3: Use Point-Slope Form

We can use the point-slope form of a line, which is \[ y - y_1 = m(x - x_1) \].Choose one of the points, say \((1,5)\), and use the slope \(m=2\) we found:\[ y - 5 = 2(x - 1) \].

Step 4: Simplify to Slope-Intercept Form

Now, let's simplify to get it into the slope-intercept form \(y = mx + b\):Start with \[ y - 5 = 2(x - 1) \]Distribute the 2:\[ y - 5 = 2x - 2 \]Add 5 to both sides:\[ y = 2x + 3 \]This is the linear equation in slope-intercept form.

Key Concepts

Slope Calculation

Calculating the slope is a fundamental step when dealing with linear equations. The slope, often represented by \( m \), measures the steepness or incline of a line. It is typically expressed as the ratio of the change in the vertical direction (the rise) to the change in the horizontal direction (the run) between two points on a line. This can be calculated using the formula:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

Here, \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the two given points. In this exercise, we are given the points \((1, 5)\) and \((4, 11)\). Using the formula:

• \( m = \frac{11 - 5}{4 - 1} = \frac{6}{3} = 2 \)

This tells us that for every 1 unit we move horizontally, we move 2 units vertically. Slope can indicate if a line is rising, falling, flat, or vertical:

• If \( m > 0 \), the line rises as it moves from left to right.
• If \( m < 0 \), the line falls as it moves from left to right.
• If \( m = 0 \), the line is flat.
• If the line is vertical, slope is undefined.

Point-Slope Form

The point-slope form of a linear equation is particularly useful when you know a line's slope and a point through which it passes. The formula is:

• \( y - y_1 = m(x - x_1) \)

This form allows you to write the equation of a line instantly when having a point \((x_1, y_1)\) and slope \(m\). In our example, using the point \((1, 5)\) and the slope \(m = 2\), we substitute these values into the point-slope formula:

• \( y - 5 = 2(x - 1) \)

This equation succinctly describes the line, leveraging both a specific point and the slope. The point-slope form is often a stepping stone to converting equations into other forms like the slope-intercept form.

Slope-Intercept Form

The slope-intercept form is another way to write the equation of a line, making the relationship between the slope and the y-intercept explicit. The general form is:

• \( y = mx + b \)

Where \( m \) represents the slope, and \( b \) is the y-intercept of the line (where the line intersects the y-axis). Converting an equation from point-slope to slope-intercept form is straightforward. Starting from:

• \( y - 5 = 2(x - 1) \)
• Distribute the slope: \( y - 5 = 2x - 2 \)
• Then, solve for \( y \) by adding 5 to both sides: \( y = 2x + 3 \)

This makes it clear that the slope of the line is 2, and it crosses the y-axis at 3. This form is especially useful for quickly graphing a line and understanding its behavior.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves placing geometric figures in a coordinate plane and using algebra to solve geometric problems. This fusion of algebra and geometry allows a robust analysis of lines, including understanding their properties and how they interact with one another on a graph.In the realm of linear equations, you will often represent lines in a Cartesian plane with x and y coordinates. Here’s how it works:

• The xy-plane is divided into four quadrants by the x-axis (horizontal) and the y-axis (vertical).
• Points on this plane are denoted by coordinates \((x, y)\).

For a linear equation like \( y = 2x + 3\), plotting this involves placing the y-intercept (\(b = 3\)) on the y-axis and then using the slope (\(m = 2\)) to determine other points:

• Start at \( (0, 3) \)
• Move 1 unit right (positive x-direction) and 2 units up (since the slope is 2) to reach the next point \((1, 5)\).

This systematic way of plotting helps visualize how the linear equation behaves, providing insight into the relationship between any two points on its line.