Problem 18 - Algebra 1

Question

Write an equation of the line in slope-intercept form. The slope is \(-6 ;\) the \(y\) -intercept is \(\frac{3}{4}\)

Step-by-Step Solution

Answer

The equation of the line in slope-intercept form is \(y = -6x + \frac{3}{4}\).

Step 1: Transcribe the slope-intercept equation

The student should first write down the general form of the slope-intercept equation which is \(y = mx + c\).

Step 2: Substitute the given values

Substitute the given slope, \(m = -6\), and the y-intercept, \(c = \frac{3}{4}\), into the slope-intercept equation. This becomes \(y = -6x + \frac{3}{4}\).

Step 3: Writing the Final Equation

After substitution, the equation of the line in slope-intercept form is \(y = -6x + \frac{3}{4}\).

Key Concepts

Slope of a Line

Understanding the slope of a line is crucial in algebra and geometry. The slope indicates the steepness and direction of a line. Mathematically, it's defined as the ratio of the rise over the run between two points on a line.

In simpler terms, if you take two points on a line, say Point A \( (x_1, y_1) \) and Point B \( (x_2, y_2) \) the slope \( m \) is calculated by the difference in the y-values divided by the difference in the x-values. The formula for this is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). For a positive slope, the line rises as it moves from left to right. Conversely, for a negative slope, the line falls. If the slope is zero, the line is horizontal which indicates no rise or fall.

Y-Intercept

The y-intercept of a line is the point where the line crosses the y-axis. It is where the x-value is zero. In the slope-intercept form of a linear equation \( y = mx + b \), \( b \) represents the y-intercept. This is an important concept as it gives us a starting point for graphing the line and understanding its position in the coordinate system.

For instance, if a line has a y-intercept of \( \frac{3}{4} \), it means that the line crosses the y-axis at the point \( (0, \frac{3}{4}) \). The slope-intercept form makes it straightforward to identify the y-intercept since it is explicitly stated as the constant term.

Writing Linear Equations

Writing linear equations is a foundational skill for graphing and interpreting linear functions. A linear equation in the slope-intercept form is written as \( y = mx + b \), where \( m \) is the slope, and \( b \) is the y-intercept.

To write an equation, you need to know these two pieces of information. Once you have the slope and the y-intercept, you simply substitute them into the slope-intercept form. This direct substitution method is a reliable way to write the equation of a line accurately. When given the slope and y-intercept, no additional transformations are needed, making it a particularly convenient form for quickly plotting lines on a graph.

Slope-Intercept Equation Substitution

Once you have the general form of the slope-intercept equation \( y = mx + b \), substituting the values to write the equation of a specific line is simple. What you're doing is replacing the variables \( m \) and \( b \) with the actual numerical values you've been given.

For example, with a slope \( m = -6 \) and a y-intercept \( b = \frac{3}{4} \), you substitute these into the equation: \( y = mx + b \) becomes \( y = (-6)x + \frac{3}{4} \). The substitution process is a practical application of algebraic skills and helps in visualizing how changes in slope and y-intercept values affect the position and steepness of the line on the graph.