Problem 12 - Understanding Elementary Algebra with Geometry

Question

Write an equation of the line satisfying the given conditions. Passing through \((0,-4)\) with slope \(-\frac{2}{7}\)

Step-by-Step Solution

Answer

The equation is \(y = -\frac{2}{7}x - 4\).

Step 1: Identify Slope-Intercept Form

The general equation for a line in slope-intercept form is given by \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept.

Step 2: Substitute Given Slope

From the problem, the slope \(m\) is given as \(-\frac{2}{7}\). Substitute this value into the slope-intercept form of the equation: \(y = -\frac{2}{7}x + b\).

Step 3: Substitute the Point into the Equation

The line passes through the point \((0, -4)\), meaning when \(x = 0\), \(y = -4\). Substitute these values into the equation: \(-4 = -\frac{2}{7}(0) + b\).

Step 4: Solve for y-intercept, b

Simplify the equation: \(-4 = b\). So, the y-intercept \(b\) is \(-4\).

Step 5: Write the Final Equation

Now that both the slope \(m\) and y-intercept \(b\) are known, write the final equation of the line: \(y = -\frac{2}{7}x - 4\).

Key Concepts

linear equations

Linear equations are equations that form a straight line when graphed on a coordinate plane. They can be written in various forms, but one of the most common is the slope-intercept form: \(y = mx + b\). Here, \(m\) represents the slope of the line, and \(b\) is the y-intercept, which is the point where the line crosses the y-axis.
Linear equations are essential because they describe relationships between variables that have a constant rate of change. In other words, as one variable increases or decreases, the other variable changes proportionately. For example, if you know the equation of a line, you can easily find out the value of \(y\) for any given \(x\) and vice versa. This characteristic makes linear equations incredibly useful for making predictions and solving real-world problems.

slope

The slope of a line, denoted as \(m\), is a measure of how steep the line is. It tells you the rate at which \(y\) changes for a given change in \(x\). Mathematically, the slope is calculated as the ratio of the rise (the change in \(y\)) to the run (the change in \(x\)): \[ m = \frac{\text{rise}}{\text{run}} \].
Slope can be positive, negative, zero, or undefined:

• A positive slope means the line inclines upwards as it moves from left to right.
• A negative slope means the line declines downwards as it moves from left to right.
• A slope of zero means the line is horizontal.
• An undefined slope means the line is vertical.

In our original exercise, the slope is given as \(-\frac{2}{7}\), which tells us that for every 7 units we move to the right, the line falls by 2 units.

y-intercept

The y-intercept, denoted as \(b\), is the point where the line crosses the y-axis of the coordinate plane. This is the value of \(y\) when \(x = 0\). To find the y-intercept in the slope-intercept form of a linear equation \(y = mx + b\), substitute \(x=0\) and solve for \(y\).
In the given problem, we are told that the line passes through the point \((0, -4)\). Therefore, substituting \(x = 0\) into our equation simplifies it to \(-4 = b\), revealing that the y-intercept \(b\) is -4.

coordinate geometry

Coordinate geometry allows us to describe the position of points on a plane using a pair of numerical coordinates \((x, y)\). This system is incredibly useful for graphing linear equations and understanding their behavior.
In a coordinate plane:

• The x-axis is the horizontal axis.
• The y-axis is the vertical axis.
• Any point in the plane can be represented as \((x, y)\).

For example, the line in our problem passes through the point \(0, -4\), which is located on the y-axis. By understanding the coordinates and how they relate to equations, we can draw accurate graphs and solve geometric problems efficiently.