Question

Line \(\ell\) has a slope of \(\frac{1}{2}\) and a \(y\)-intercept of \(-7\). Which of the following is the equation of \(\ell\) written in slopeintercept form? (A) \(y=\frac{1}{2} x-7\) (B) \(y=\frac{1}{2} x+7\) (C) \(2 x+y=7\) (D) \(x-2 y=14\)

Short answer

The equation is \( y = \frac{1}{2} x - 7 \) (Option A).

Step-by-step solution

Understanding the Slope-Intercept Form

The slope-intercept form of a line is given by the equation \( y = mx + c \), where \( m \) represents the slope of the line, and \( c \) represents the \( y \)-intercept.

Identifying Given Information

We are given that the slope \( m \) is \( \frac{1}{2} \) and the \( y \)-intercept \( c \) is \(-7\).

Substitute Values into Slope-Intercept Form

Substitute \( m = \frac{1}{2} \) and \( c = -7 \) into the slope-intercept form equation: \( y = \frac{1}{2}x - 7 \).

Matching with Given Options

Compare the formed equation \( y = \frac{1}{2}x - 7 \) with the options. Option (A) is exactly the same as this equation, so it is the correct choice.

Key concepts

linear equations

A linear equation is a mathematical expression that represents a straight line when plotted on a graph. The hallmark of a linear equation is its constant slope and straight-line appearance when graphed.
Linear equations can be expressed in several forms, but one of the most common and useful is the slope-intercept form. This form is particularly helpful when you want to quickly ascertain the slope and y-intercept of a line.
When working with linear equations:

• Each term is either a constant or a product of a constant and an independent variable like \(x\) or \(y\).
• The highest power of the variable is always 1, characteristic of linear equations.
• You can rewrite linear equations in different forms, making them versatile in solving a wide variety of problems.

Understanding the structure and properties of linear equations is essential, especially when determining a line's features. They form the basis for more complex algebraic concepts, making them integral to core mathematical learning.

slope

The slope of a line, denoted as \(m\) in the slope-intercept form, is a measure of how steep the line is. Essentially, it tells us how much the line goes up or down as we move along it.
The concept of slope is crucial in understanding and interpreting linear relationships.

• The slope is calculated as the "rise" over the "run," or how much the line goes up or down for each step to the right.
• A positive slope means that the line ascends as you move right, while a negative slope indicates it descends.
• A slope of zero indicates a perfectly horizontal line, showing no vertical change as you move horizontally.
• The larger the absolute value of the slope, the steeper the line.

In our exercise, the slope is \(\frac{1}{2}\), which means for every 1 unit the line moves right, it rises by 0.5 units. This gently sloping line indicates a moderate increase as \(x\) increases. Being able to identify and interpret slopes is essential for graphing and understanding the behavior of linear equations.

y-intercept

The y-intercept, represented as \(c\) in the slope-intercept form of a line, corresponds to the point where the line intersects the y-axis. This value is fundamental because it provides a starting point from which to graph the line.
The y-intercept gives valuable information without needing to calculate or graph much else.

• It is the value of \(y\) when \(x = 0\), making it easy to find on the y-axis.
• The y-intercept can tell you instantly where a line crosses the vertical axis on a graph, which is helpful when plotting the line.
• A positive y-intercept means the line crosses the y-axis above the origin, while a negative y-intercept, like our example of -7, means it crosses below.

In the provided problem, the y-intercept is -7, indicating that the line crosses the y-axis 7 units below the origin. Recognizing this point aids significantly in quickly drawing the graph of the linear equation and understanding its behavior.