Problem 3 - Understanding Elementary Algebra with Geometry

Question

Write an equation of the line satisfying the given conditions. Passing through \((-1,4)\) with slope \(\frac{1}{2}\)

Step-by-Step Solution

Answer

The equation of the line is \( y = \frac{1}{2}x + \frac{9}{2} \).

Step 1 - Understand the Slope-Intercept Form

The slope-intercept form of a line's equation is given by \[ y = mx + b \] where \( m \) is the slope and \( b \) is the y-intercept.

Step 2 - Identify Given Values

We are given a point \((-1, 4)\) and a slope \( \frac{1}{2} \). So in our equation, \( m = \frac{1}{2} \) and the point satisfies \( (x, y) = (-1, 4) \).

Step 3 - Substitute Values into the Equation

Substitute the slope \( m \) and the coordinates of the point into the equation \[ y = mx + b \] which gives us \[ 4 = \frac{1}{2}(-1) + b \].

Step 4 - Solve for b

Solve the equation \[ 4 = \frac{1}{2}(-1) + b \] which simplifies to \[ 4 = -\frac{1}{2} + b \]. Add \( \frac{1}{2} \) to both sides to isolate \( b \): \[ 4 + \frac{1}{2} = b \]. This means \[ b = \frac{9}{2} \].

Step 5 - Write the Final Equation

Substitute \( m = \frac{1}{2} \) and \( b = \frac{9}{2} \) back into the slope-intercept form equation: \[ y = \frac{1}{2}x + \frac{9}{2} \].

Key Concepts

slope-intercept form

The slope-intercept form of a line is a popular way to represent linear equations. This form is written as:
\[ y = mx + b \]
Here, the variable y represents the y-coordinate, and x represents the x-coordinate of any point on the line. The value m is the slope of the line, which indicates the steepness or tilt of the line. The value b is the y-intercept, which is the point where the line crosses the y-axis.
Understanding these two components, the slope and the y-intercept, helps in quickly graphing or interpreting the behavior of the line.

slope

The slope (\( m \)) of a line measures how steep the line is. You can think of it as the rise over run which describes how much y changes for a given change in x. Symbolically, slope is written as:
\[ m = \frac{{\text{rise}}}{{\text{run}}} \]
In simpler terms, it's the amount of vertical change per unit of horizontal change. In our example, the slope is given as \( \frac{1}{2} \), which means for every 2 units you move to the right (run), the line rises by 1 unit (rise).
Knowing the slope helps in determining if the line is increasing (positive slope) or decreasing (negative slope).

y-intercept

The y-intercept (\( b \)) is where the line crosses the y-axis. In other words, it's the value of y when x is zero. When we solve for \( b \), we set x to zero and find out where the line hits the y-axis.
In the problem, after substituting the slope (\( \frac{1}{2} \)) and the given point ((-1, 4) ), we solved for b:
\[ 4 = -\frac{1}{2} + b \]
By isolating \( b \), we found that:
\[ b = \frac{9}{2} \]
This value indicates the precise point on the y-axis where the line intersects. This makes plotting the line straightforward since you now know one exact point it crosses (0, \( \frac{9}{2} \)).