Problem 5 - Understanding Elementary Algebra with Geometry

Question

Write an equation of the line satisfying the given conditions. Passing through \((-3,-5)\) with slope \(-\frac{2}{3}\)

Step-by-Step Solution

Answer

The equation of the line is \( y = -\frac{2}{3}x - 7 \)

Step 1: Identify the Point-Slope Form of the Line

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

Step 2: Substitute the Point and Slope into the Equation

Substitute the given point \( (-3, -5) \) and slope \( m = -\frac{2}{3} \) into the point-slope form equation: \[ y - (-5) = -\frac{2}{3}(x - (-3)) \] Simplify to get: \[ y + 5 = -\frac{2}{3}(x + 3) \]

Step 3: Distribute the Slope on the Right Side

Distribute \( -\frac{2}{3} \) through the expression \( (x + 3) \): \[ y + 5 = -\frac{2}{3}x - 2 \]

Step 4: Isolate the Variable y

Subtract 5 from both sides of the equation to isolate \( y \): \[ y = -\frac{2}{3}x - 2 - 5 \] Simplify the constants on the right: \[ y = -\frac{2}{3}x - 7 \]

Key Concepts

Point-Slope Form

The point-slope form of a linear equation is a useful way to write the equation of a line when you know a point on the line and the line's slope. It is represented as: \[ y - y_1 = m(x - x_1) \]In this formula,

• \((x_1, y_1)\) is a specific point on the line, with \(x_1\) as the x-coordinate and \(y_1\) as the y-coordinate.
• The letter \(m\) represents the slope of the line.

By using this form, you can easily plug in the known slope and point to get the equation of the line. This form is handy because it directly uses the slope (pointing at how steep the line is) and a specific point, making it simple to translate real-world problems into linear equations.

Slope

The slope of a line is a measure of how steep the line is. It is often represented by the letter m. Mathematically, slope is calculated as the 'rise' (change in y) over the 'run' (change in x). The formula for slope is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Here, the slope m tells you how much y changes for a given change in x.

• A positive slope means the line rises as it moves from left to right.
• A negative slope means the line falls as it moves from left to right.
• A slope of zero means the line is horizontal.
• An undefined slope means the line is vertical and cannot be expressed as a single number.

In our exercise, the slope is given as -\frac{2}{3}, meaning for every 3 units the line moves horizontally to the right, it moves down by 2 units.

Linear Equations

A linear equation is an equation that makes a straight line when graphed. The standard form of a linear equation is: \[ Ax + By = C \]Where A, B, and C are constants. Another common form is the slope-intercept form: \[ y = mx + b \]Here, m represents the slope, and b represents the y-intercept (the point where the line crosses the y-axis). Linear equations can describe many relationships in real life, such as calculating speed, predicting costs, and understanding trends. In the original exercise, our final linear equation was: \[ y = -\frac{2}{3}x - 7 \]This equation tells us the slope is -\frac{2}{3} and the y-intercept is -7. Therefore, every time you use this equation, you know the line crosses the y-axis at -7 and has a slope of -\frac{2}{3}.