Problem 12 - Algebra 1: Concepts and Skills

Question

Write in slope-intercept form the equation of the line described below. Slope \(=14, y\) -intercept \(=-6\)

Step-by-Step Solution

Answer

The equation of the line in slope-intercept form is \(y=14x-6\).

Step 1: Identify slope and y-intercept

The exercise directly presents the slope (m) as 14 and the y-intercept (b) as -6.

Step 2: Substitute slope and y-intercept into the slope-intercept form

Substitute the given slope and y-intercept values into the slope-intercept form equation \(y=mx+b\). So, with \(m=14\) and \(b=-6\), the equation becomes \(y=14x-6\).

Step 3: Final Equation

The final equation in slope-intercept form is \(y=14x-6\). No further steps are required as this is the final answer.

Key Concepts

Linear Equations

A linear equation is a powerful tool in mathematics, most commonly represented on a graph as a straight line. It takes the form of an equation like:

• Standard Form: \(Ax + By = C\)
• Slope-Intercept Form: \(y = mx + b\)

In both forms, the equation describes a straight path, with each point on the line being a solution to the equation. This relationship implies that a change in one variable results in a predictable change in the other.

The beauty of linear equations lies in their simplicity and versatility. In essence, they provide a straightforward way to relate two quantities that are proportionally linked. In many real-world applications, such as predicting expenses, motion, or growth trends, linear equations serve as excellent approximations for understanding these relationships.

Y-intercept

The y-intercept is a crucial component of a linear equation. In the slope-intercept form \(y = mx + b\), the y-intercept is represented by \(b\). This value indicates where the line crosses the y-axis on a graph.

Imagine the graph as a physical, two-dimensional space. The y-intercept tells us the starting point of the line when \(x = 0\).

• If the y-intercept is positive, the line crosses the y-axis above the origin.
• If it's negative, it crosses below the origin.

Understanding the y-intercept helps you quickly visualize where a line is positioned relative to the origin, making it a foundational concept in graphing linear equations. In our problem, the y-intercept is \(-6\), meaning the line meets the y-axis 6 units below the origin.

Slope

The slope of a line indicates its steepness and direction. In the equation \(y = mx + b\), \(m\) is the slope.

• A positive slope means the line rises as it moves from left to right.
• A negative slope means the line falls.
• The greater the slope's absolute value, the steeper the line.

The slope is calculated as the change in \(y\) divided by the change in \(x\) between two points on the line (known as "rise over run"). Mathematically, it's expressed as \(m = \frac{\Delta y}{\Delta x}\).

Understanding the slope lets us predict how a line changes and makes it easier to graph. For instance, with a slope of \(14\), as in our example, the line is steep and rises sharply as \(x\) increases.