Problem 40 - Algebra and Trigonometry

Question

(a) Sketch the line with slope \(-2\) that passes through the point \((4,-1)\) . (b) Find an equation for this line.

Step-by-Step Solution

Answer

The line has equation \( y = -2x + 7 \).

Step 1: Understanding the Slope-Point Form of a Line

The slope-point form of a line is useful when we know a point on the line and the slope of the line. It is given by the formula \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \((x_1, y_1)\) is a point on the line.

Step 2: Identify the Known Values

From the problem, we know the slope \( m = -2 \), and the point \((x_1, y_1)\) through which the line passes is \((4, -1)\).

Step 3: Plug the Known Values into the Slope-Point Form

Substitute \( m = -2 \), \( x_1 = 4 \), and \( y_1 = -1 \) into the equation \( y - y_1 = m(x - x_1) \). This gives us \( y - (-1) = -2(x - 4) \).

Step 4: Simplify the Equation

Simplify the expression to put it in the slope-intercept form \( y = mx + b \). Start by simplifying \( y + 1 = -2(x - 4) \). Distributing the \(-2\) gives \( y + 1 = -2x + 8 \). Then, solve for \( y \) to get \( y = -2x + 7 \).

Step 5: Sketch the Line

To sketch the line, start at the point \((4, -1)\) on a coordinate plane. Use the slope, which is \(-2\), to determine the direction and steepness of the line: For every unit you move to the right along the x-axis, move 2 units down on the y-axis, since the slope is negative. Draw the line accordingly.

Key Concepts

Slope-Point Form

The slope-point form of an equation of a line is an essential tool for defining lines on a coordinate plane. This form is particularly helpful when you know one point on the line and the slope of the line. The formula for the slope-point form is:

• \( y - y_1 = m(x - x_1) \)

Here, \( m \) represents the slope of the line, and \((x_1, y_1)\) denotes a specific point on the line where the coordinates \( x_1 \) and \( y_1 \) are known. It's a straightforward way to visualize and sketch the line, ensuring the equation perfectly aligns with the given point and slope.
When using this form, always substitute the known values directly into the equation. For instance, if you have a slope of \(-2\) and a point \((4, -1)\), substitute them into the formula to get:

• \( y - (-1) = -2(x - 4) \)

This input helps in shaping the understanding of the relationship between the slope and the coordinate point.

Slope-Intercept Form

Once you have an equation in slope-point form, it's often useful to convert it into the slope-intercept form. This form is popular for its simplicity and clarity, showing both the slope and the y-intercept directly. The formula for slope-intercept form is:

• \( y = mx + b \)

In this equation, \( m \) again represents the slope, while \( b \) is the y-intercept, the point where the line crosses the y-axis.
To convert an equation from slope-point to slope-intercept form, expand and simplify the equation. Taking our example of \( y + 1 = -2(x - 4) \), distribute the slope to get:

• \( y + 1 = -2x + 8 \)

Then isolate \( y \) on one side to find:

• \( y = -2x + 7 \)

Here, \(-2\) is the slope, and \(7\) is the y-intercept. This form makes it easy to visualize and graph the line on a coordinate plane.

Coordinate Plane

The coordinate plane is a fundamental concept in mathematics, serving as a space where we can graph equations and visualize relationships between variables. It is a two-dimensional surface divided by a horizontal line (x-axis) and a vertical line (y-axis), intersecting at a point called the origin \((0,0)\).
When sketching a line using the coordinates given by an equation, start by plotting specific points. For example, with a known point on the line \((4, -1)\), place this point on the coordinate plane to begin your drawing.

• The x-axis represents all possible values for \( x \).
• The y-axis shows all values for \( y \).

Understanding the slope is crucial. It tells us how far and in what direction to move from one point to another. If the slope is \(-2\), meaning for each step right on the x-axis, move two steps down along the y-axis, you can continue plotting more points and draw a straight line through them.
This is how you visualize equations like \( y = -2x + 7 \) on the coordinate plane, reflecting all the underlying relationships between the algebraic expressions and their geometric representations.