Problem 11 - Understanding Elementary Algebra with Geometry

Question

Write an equation of the line satisfying the given conditions. Passing through \((-2,0)\) with slope \(-\frac{3}{4}\)

Step-by-Step Solution

Answer

The equation of the line is \( y = -\frac{3}{4}x - \frac{3}{2} \).

Step 1: Understand the point-slope form

Recall the point-slope form of a line equation, which is given by: \[ y - y_1 = m(x - x_1) \] where \( (x_1, y_1) \) is a point on the line and \( m \) is the slope.

Step 2: Plug in the given point and slope

Plug in the point \((-2, 0)\) and slope \(-\frac{3}{4}\) into the point-slope form equation. Here, \( x_1 = -2 \), \( y_1 = 0 \), and \( m = -\frac{3}{4} \).\[ y - 0 = -\frac{3}{4}(x - (-2)) \]

Step 3: Simplify the equation

Simplify the equation from Step 2:\[ y = -\frac{3}{4}(x + 2) \]Distribute the slope:\[ y = -\frac{3}{4}x - \frac{3}{4} \times 2 \]\[ y = -\frac{3}{4}x - \frac{3}{2} \]

Key Concepts

Point-Slope Form

In linear equations, the point-slope form is essential. It helps in writing the equation of a line when you know a point and the slope. The formula is:

• y - y_1 = m(x - x_1)

Here, \( m \) stands for the slope, and \( (x_1, y_1) \) represents a point on the line. For example, if a line passes through \( (-2, 0) \) and has a slope of \( -\frac{3}{4} \), you can plug these values into the formula.

• When \( x_1 = -2 \) and \( y_1 = 0 \), the equation becomes: \( y - 0 = -\frac{3}{4}(x - (-2)) \).
• This simplifies to \( y = -\frac{3}{4}(x + 2) \).

Point-slope form is particularly handy when you transform the equation into other formats, such as slope-intercept form.

Slope-Intercept Form

The slope-intercept form is another crucial representation of linear equations. This form makes it easy to identify the slope and y-intercept of a line. Its general formula is:ul>

y = mx + b

Here, \( m \) is the slope, and \( b \) is the y-intercept. Let's convert the point-slope form equation \( y = -\frac{3}{4}(x + 2) \) into the slope-intercept form.

• First, distribute the slope: \( y = -\frac{3}{4}x - \frac{3}{2} \).
• In this equation, \( -\frac{3}{4} \) is the slope, and \( -\frac{3}{2} \) is the y-intercept.

This format is useful for quickly graphing the line and understanding its behavior.

Coordinate Geometry

Coordinate geometry connects algebra and geometry through graphs of equations. It's the study of geometric figures using the coordinate plane. Here, we find points based on their coordinates (x and y values). For instance, the point \( (-2, 0) \) in our exercise has its x-coordinate as -2 and y-coordinate as 0.

• Using the point-slope form, we find how the line behaves around this point.
• Slope helps determine how steep the line is and whether it goes up or down as you move along the x-axis.

Coordinate geometry makes it easier to visualize equations and understand the spatial relationships between points, lines, and curves. It's essential for graphing and solving geometric problems algebraically.