Problem 41 - Algebra 2

Question

What is an equation of the line through \(\left(\frac{1}{2},-\frac{3}{2}\right)\) and \(\left(-\frac{1}{2}, \frac{1}{2}\right) ?\) A. \(y=-2 x-\frac{1}{2}\) B. \(y=-3 x\) C. \(y=2 x-\frac{5}{2}\) D. \(y=\frac{1}{2} x+1\)

Step-by-Step Solution

Answer

The equation of the line is \(y = -2x - \frac{1}{2}\) (option A).

Step 1: Find the Slope

To find the slope (m) of the line passing through points \(\left(\frac{1}{2}, -\frac{3}{2}\right)\) and \(\left(-\frac{1}{2}, \frac{1}{2}\right)\), we use the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substituting the coordinates, we get \[m = \frac{\frac{1}{2} + \frac{3}{2}}{-\frac{1}{2} - \frac{1}{2}} = \frac{2}{-1} = -2.\] The slope \(m = -2.\)

Step 2: Use Point-Slope Form

With the slope \(m = -2\) and a point \((\frac{1}{2}, -\frac{3}{2})\) on the line, we apply the point-slope form of the equation: \(y - y_1 = m(x - x_1)\). Substituting, we have \[y + \frac{3}{2} = -2(x - \frac{1}{2}).\]

Step 3: Simplify the Equation

Expand and simplify the point-slope equation: \[y + \frac{3}{2} = -2x + 1.\] Move terms to isolate \(y\): \[y = -2x + 1 - \frac{3}{2} = -2x - \frac{1}{2}.\] This matches option A.

Key Concepts

Slope Calculation

Calculating the slope of a line is a fundamental concept in coordinate geometry. The slope, often denoted as \( m \), represents the steepness and direction of a line. To find the slope of a line passing through two points, use the formula:

• \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

In this formula, \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the two points. The numerator \( y_2 - y_1 \) represents the vertical change (rise), and the denominator \( x_2 - x_1 \) represents the horizontal change (run) between the points.

When substituting the given points into the formula

• \( rac{1}{2} + \frac{3}{2} \)
• \(-\frac{1}{2} - \frac{1}{2} \)

we find \( m = -2 \), indicating the line slopes downwards as it moves from left to right.
This means for every unit you move horizontally, the line moves vertically in the opposite direction by two units.

Point-Slope Form

The point-slope form is a useful way to write the equation of a line when you know the slope and a single point on the line. The formula is expressed as:

• \[ y - y_1 = m(x - x_1) \]

Here, \( m \) is the slope, and \((x_1, y_1)\) is the known point. To find the equation of a line with the calculated slope \(-2\) and the point \((\frac{1}{2}, -\frac{3}{2})\), substitute these values into the formula:

\[ y + \frac{3}{2} = -2(x - \frac{1}{2}) \].

This setup allows you to establish an initial relationship between the \( x \) and \( y \) values that lie on this line

• Reflects an intuitive way to "fix" the line according to the known data.

The point-slope form is particularly helpful for easily adjusting and identifying new lines with different slopes or points.

Coordinate Geometry

Coordinate geometry, or analytic geometry, involves the study of geometric figures using coordinates on a plane. Each point in this space is defined by an \((x, y)\) coordinate, which allows precise positioning and measurement of figures.

In problems involving lines, coordinate geometry provides tools like distance measurement between points, slope calculation, and the ability to form linear equations. Understanding these elements is crucial because:

• Points define the specific segments of a line.
• The slope quantifies the line's steepness or inclination.
• Equations use these properties to describe the entire line.

Coordinate geometry helps visualize relationships between geometric constructs, making it easier to analyze objects like lines through established formulas. This makes solving geometry problems in a coordinate plane systematic and reliable, as it often accompanies visual aids such as graphs.

Linear Equations

Linear equations are algebraic expressions that define straight lines on a coordinate plane. These equations are crucial in mathematics and appear frequently across various applications. They generally take the form:

• \( y = mx + c \)

where \( m \) is the slope and \( c \) is the y-intercept, the point where the line crosses the y-axis.

In our exercise, starting from the point-slope form and simplifying, we arrived at the linear equation

• \( y = -2x - \frac{1}{2} \)

This standard form is useful for analyzing and predicting the behavior of the line, such as identifying where it intersects the axes or how to solve for \( y \) given an \( x \). Linear equations bridge algebra and geometry while presenting a straightforward way to depict lines algebraically.