Problem 9 - Understanding Elementary Algebra with Geometry

Question

Write an equation of the line satisfying the given conditions. Passing through \((0,-2)\) with slope \(\frac{1}{4}\)

Step-by-Step Solution

Answer

The equation of the line is \( y = \frac{1}{4}x - 2 \).

Step 1: Identify the Point-Slope Form

The point-slope form of a line equation 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: Substitute Given Values

Substitute the given slope \( m = \frac{1}{4} \) and the point \( (0, -2) \) into the point-slope form. This gives us: \[ y - (-2) = \frac{1}{4}(x - 0) \].

Step 3: Simplify the Equation

Simplify the equation to get it into a more standard form. This gives us: \[ y + 2 = \frac{1}{4}x \].

Step 4: Isolate y

Subtract 2 from both sides to solve for \( y \):\[ y = \frac{1}{4}x - 2 \].

Key Concepts

point-slope form

The point-slope form of an equation of a line is a powerful tool to use when you know a point on the line and the slope. This form of the equation is written as: \[ y - y_1 = m(x - x_1) \], where \( (x_1, y_1) \) is the known point on the line, and \( m \) is the slope. It is particularly useful because it directly incorporates these two pieces of information. For example, if you have a point at \((0, -2)\) and a slope of \( \frac{1}{4} \), you can quickly set up the equation. The clear advantage of the point-slope form lies in its simplicity and directness.

slope

The slope \( m \) of a line is a measure of how steep the line is. It can be found by the ratio of the rise (change in \( y \)) to the run (change in \( x \)). For a slope of \( \frac{1}{4} \), every time \( x \) increases by 4 units, \( y \) increases by 1 unit. The slope can be negative or positive, meaning the line can tilt upwards or downwards. Imagine a hill: a positive slope means the hill goes upward as you move from left to right, while a negative slope means it goes downward.

simplifying equations

After substituting the given values into the point-slope form, the next step is to simplify the equation. Starting from \[ y - (-2) = \frac{1}{4}(x - 0) \], you simplify to \[ y + 2 = \frac{1}{4}x \]. Simplifying equations often involves combining like terms and reducing fractions. The goal is to reach a simpler or more standard form, like slope-intercept form \( y = mx + b \). The process helps in understanding the line's behavior and making calculations easier.

solving for y

Solving for \( y \) means getting \( y \) on one side of the equation to express it in terms of \( x \). Starting from the simplified equation \[ y + 2 = \frac{1}{4}x \], you subtract 2 from both sides to isolate \( y \). This gives you: \[ y = \frac{1}{4}x - 2 \]. This is now in the slope-intercept form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Solving for \( y \) provides a clear way to see how \( y \) changes with \( x \) and makes graphing the line straightforward.