Problem 75 - College Algebra with Corequisite Support

Question

For the following exercises, sketch a line with the given features. Passing through the points (-3,-4) and (3,0)

Step-by-Step Solution

Answer

The line has equation \( y = \frac{2}{3}x - 2 \), and can be sketched using points \((-3,-4)\) and \((3,0)\).

Step 1: Identify the Slope Formula

The first step is to determine the slope of the line passing through the two points. The slope formula is given by \( m = \frac{y_2 - y_1}{x_2 - x_1} \), where \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the two points.

Step 2: Calculate the Slope

Substitute the values from the points \((-3, -4)\) and \((3, 0)\) into the slope formula: \( m = \frac{0 - (-4)}{3 - (-3)} = \frac{4}{6} = \frac{2}{3} \). So, the slope of the line is \( \frac{2}{3} \).

Step 3: Use Point-Slope Form to Find Equation

With the slope \( m = \frac{2}{3} \) and using the point-slope form of a line \( y - y_1 = m(x - x_1) \), choose one of the points to find the line equation. Using the point \((-3, -4)\), we have \( y - (-4) = \frac{2}{3}(x - (-3)) \).

Step 4: Simplify the Equation

Simplify the equation from Step 3: \( y + 4 = \frac{2}{3}(x + 3) \). Distribute the slope to obtain: \( y + 4 = \frac{2}{3}x + 2 \). Subtract 4 from both sides to isolate \( y \): \( y = \frac{2}{3}x - 2 \). This is the equation of the line in slope-intercept form.

Step 5: Sketch the Line

To sketch the line, use the two points \((-3, -4)\) and \((3, 0)\) on the coordinate plane. You know the line has a slope of \( \frac{2}{3} \), meaning it rises 2 units for every 3 units it runs to the right. Use the equation and slope to verify the accuracy of your line.

Key Concepts

Slope Formula

Understanding the slope of a line is crucial in mathematics. The slope formula is a simple, yet powerful tool that helps us identify the steepness of a line. It tells us how many units the line goes up (or down) for every unit it goes right. The formula is expressed as:

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

Here, \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of two points on the line. The slope \(m\) is the "rise" (change in \(y\)) divided by the "run" (change in \(x\)). To see this in action, let's consider the points \((-3, -4)\) and \((3, 0)\). By plugging these values into the formula, we calculate the slope as follows:

• \( m = \frac{0 - (-4)}{3 - (-3)} = \frac{4}{6} = \frac{2}{3} \)

This tells us the line rises 2 units for every 3 units it moves horizontally.

Point-Slope Form

Once the slope is calculated, we can find the equation of the line using the point-slope form. This form is particularly useful when we know the slope and a single point on the line. The equation is structured as:

• \( y - y_1 = m(x - x_1) \)

Here, \(m\) is the slope and \((x_1, y_1)\) is the known point. Using the slope \( \frac{2}{3} \) and the point \((-3, -4)\), the equation becomes:

• \( y - (-4) = \frac{2}{3}(x - (-3)) \)

Simplifying, we have:

• \( y + 4 = \frac{2}{3}(x + 3) \)

This step sets the foundation for converting the equation into a more manageable format.

Slope-Intercept Form

The slope-intercept form is a convenient way to express the equation of a line. It makes it easy to identify both the slope and the y-intercept directly from the equation. The formula is:

• \( y = mx + b \)

Here, \(m\) is the slope and \(b\) is the y-intercept, which is where the line crosses the y-axis. From the point-slope equation \( y + 4 = \frac{2}{3}(x + 3) \), we simplify to reach the slope-intercept form. Distribute the \(\frac{2}{3}\) through the parentheses:

• \( y + 4 = \frac{2}{3}x + 2 \)

Then, subtract 4 from both sides to isolate \(y\):

• \( y = \frac{2}{3}x - 2 \)

Now, it’s clear that the slope is \(\frac{2}{3}\) and the y-intercept is -2.

Coordinate Plane

The coordinate plane is a fundamental concept in geometry and algebra. It's a two-dimensional surface made up of a horizontal axis (x-axis) and a vertical axis (y-axis). Points on this plane are identified by their coordinates \((x, y)\).When sketching a line like the one from our exercise, the coordinate plane comes in handy. Plot the given points \((-3, -4)\) and \((3, 0)\) on the plane. These points will guide where the line should be placed.Knowing the line's slope \(\frac{2}{3}\) tells us how the line moves between these points. It rises 2 units for every 3 units it runs to the right. Thus, the coordinate plane not only helps in placing the line but also in visualizing its direction and steepness.