Problem 10 - Understanding Elementary Algebra with Geometry

Question

Write an equation of the line satisfying the given conditions. Passing through \((-4,0)\) with slope \(\frac{1}{5}\)

Step-by-Step Solution

Answer

y = \frac{1}{5}x + \frac{4}{5}

Step 1: Identify the point-slope form

To write the equation of a line, use the point-slope form of the equation: \[ y - y_1 = m (x - x_1) \]where \( (x_1, y_1) \) is a point on the line and \( m \) is the slope.

Step 2: Substitute the given point and slope

Substitute the given point \( (-4,0) \) and slope \( \frac{1}{5} \) into the point-slope form: \[ y - 0 = \frac{1}{5} (x + 4) \]

Step 3: Simplify the equation

Simplify the equation to get it into a more familiar format. Start by distributing the slope on the right-hand side:\[ y = \frac{1}{5} x + \frac{4}{5} \]

Key Concepts

Point-Slope Form

When dealing with linear equations, one of the most useful forms you'll encounter is the point-slope form. This is particularly helpful when you know a specific point on the line and the slope. The point-slope form of an equation is given by:
\[ y - y_1 = m (x - x_1) \] Here, \'m\' represents the slope of the line, while \'(x_1, y_1)\' is a point on the line.
As an example, if you're given the point \( (-4, 0) \) and the slope \( \frac{1}{5} \), you can substitute these values to get:
\[ y - 0 = \frac{1}{5} (x + 4) \]
Understanding this form helps in quickly drafting the equation of the line. Remember, this format is quite flexible. Altering variables takes minimal effort!

Slope-Intercept Form

Another essential form of linear equations is the slope-intercept form. This form is popular because of its straightforward interpretation of the slope and y-intercept. The slope-intercept form is written as:
\[ y = mx + b \]
In this equation, \'m\' stands for the slope (rate of change) and \'b\' is the y-intercept—the point where the line crosses the y-axis.
Continuing with our earlier example, after substituting the given point and slope into the point-slope form, and simplifying, we get:
\[ y = \frac{1}{5} x + \frac{4}{5} \]
This transformed equation is now in the slope-intercept form. It shows that the slope is \( \frac{1}{5} \) and the line intersects the y-axis at \( \frac{4}{5} \).
Slope-intercept form is quite useful for graphing and understanding the relationship between variables.

Linear Equations

Linear equations are fundamental in algebra and appear frequently in various real-world applications. They describe a relationship where change happens at a constant rate. The general form of a linear equation is:
\[ Ax + By = C \]
Linear equations can be transformed into different forms depending on the available information. If you know a point and the slope, the point-slope form is handy. For easy graphing and seeing the slope directly along with the y-intercept, converting it to the slope-intercept form is beneficial.
For instance, considering our previous solution starting from the point-slope form and simplifying to the slope-intercept form, we had:
\[ y = \frac{1}{5} x + \frac{4}{5} \]
This equation graphically represents a straight line with a constant rate of change.
Being familiar with different forms helps in recognizing and solving various problems involving linear relationships.