Problem 29 - College Algebra and Calculus: An Applied Approach

Question

Find an equation of the line that passes through the point and has the indicated slope. Then sketch the line. Point \(\quad\) Slope \((-3,6)\) \(m=-2\)

Step-by-Step Solution

Answer

The equation of the line that passes through the point (-3,6) and has the slope -2 is \(y = -2x\). The line can be sketched as a straight decreasing line that passes through the origin.

Step 1: Identify the given point and slope

The point that the line passes through is (-3, 6) and the slope \(m\) of the line is -2.

Step 2: Use the point-slope form

Insert the given point (-3,6) and slope -2 into the point-slope form \(y - y1 = m(x - x1)\) to get the equation of the line. Hence, the equation becomes \(y - 6 = -2(x - (-3))\).

Step 3: Simplify the equation

Simplify \(y – 6 = -2(x - (-3))\) to get \(y = -2x - 6 + 6\). This further simplifies to \(y = -2x\).

Step 4: Sketching the line

The line should be sketched on a grid where the y-intercept is at 0 (because no constant is in the equation), decrease 2 units downwards for every one unit moved to the right.

Key Concepts

Point-Slope Form

The point-slope form is a useful way to write the equation of a line when you know a point on the line and the slope. The formula for point-slope form is:
\[ y - y_1 = m(x - x_1) \]

• \(y\) and \(x\) represent the variables in our equation.
• \(m\) is the slope of the line.
• \((x_1, y_1)\) refers to the coordinates of the given point on the line.

By substituting the known values into this form, we can quickly find the linear equation that describes our line. In the exercise, the point given is \((-3, 6)\) and the slope \(m\) is -2. Using these values in the point-slope equation, we have:
\[ y - 6 = -2(x + 3) \].This equation represents the line passing through the specified point with the given slope.

Slope

The concept of a slope is fundamental to understanding how lines behave on a graph. It measures the steepness or tilt of a line and is usually denoted by the symbol \(m\). To calculate the slope, you'd use the formula:
\[ m = \frac{\Delta y}{\Delta x} \]

• \(\Delta y\) is the change in the \(y\) values.
• \(\Delta x\) is the change in the \(x\) values.

In simpler terms, slope is the amount by which \(y\) increases or decreases when \(x\) increases by one unit. In the given problem, the slope is \(-2\), meaning that for every unit you move right along the x-axis, the line falls by two units on the y-axis. Understanding the slope helps in quickly sketching the line, as it directly influences the direction and angle of the line on a graph.

Graphing Lines

Graphing lines is an essential skill that visually represents linear equations. When graphing a line, the equation can be thought of as instructions on how to draw it. Here's a step-by-step guide to help you graph lines:

• **Determine the y-intercept:** The y-intercept is where the line crosses the y-axis. In the equation \(y = mx + b\), the y-intercept \(b\) is the constant term. In this case, for the equation \(y = -2x\), the y-intercept is 0.
• **Plot the y-intercept:** In this example, you would plot the point \((0, 0)\) on the graph.
• **Use the slope to find another point:** From the y-intercept, use the slope to locate another point. Since the slope \(m = -2\), move 1 unit to the right and 2 units down (because the slope is negative), leading you to the point \((1, -2)\).
• **Draw the line:** Connect these points using a straight line. This line is the visual representation of the equation \(y = -2x\).

Graphing lines helps you quickly understand the relationship between \(x\) and \(y\), and it's essential for visualizing solutions to linear equations.