Problem 45 - Algebra 1

Question

Write an equation of the line in slope-intercept form. The slope is \(\frac{1}{2} ;\) the \(y\) -intercept is \(-8\).

Step-by-Step Solution

Answer

The equation of the line in slope-intercept form is \(y = \frac{1}{2}x - 8\).

Step 1: Identify the given slope and y-intercept

The problem has provided slope (m) as \(\frac{1}{2}\) and y-intercept (b) as -8.

Step 2: Write down the slope-intercept form

The slope-intercept form of a line is \(y = mx + b\) where 'm' is the slope and 'b' is the y-intercept.

Step 3: Substitute the values of m and b

Substitute the given values of the slope (m = \(\frac{1}{2}\)) and the y-intercept (b = -8) into the slope-intercept equation. The equation becomes \(y = \frac{1}{2}x - 8\).

Key Concepts

Linear Equations

A linear equation is a mathematical expression that describes a straight line when graphed on a coordinate plane. It can be written in different forms, but in most high school contexts, it is often presented as the slope-intercept form: \( y = mx + b \). Here, 'y' and 'x' represent the variables of the equation, while 'm' and 'b' are constants that affect the line's position and steepness.

• The 'm' in the equation represents the slope of the line.
• The 'b' represents the y-intercept of the line.

A linear equation suggests a constant rate of change between the two variables involved, meaning as 'x' changes, 'y' will change in a predictable manner. This predictable relationship is what makes these equations "linear." Linear equations are foundational to algebra and are crucial for understanding more advanced math like calculus. They apply to various real-world scenarios such as calculating speed, predicting trends, or converting currency.

y-intercept

The y-intercept is a key concept when dealing with linear equations in the slope-intercept form. It is represented by the 'b' in the equation \( y = mx + b \). The y-intercept is the point where the line crosses the y-axis on a graph.

• In our examples, the y-intercept is -8.
• This means that when \( x = 0 \), the value of \( y \) is -8.

Knowing the y-intercept gives you a starting point for graphing the line on the coordinate plane. You simply locate the 'b' value on the y-axis and mark that point. When visually interpreting data, the y-intercept can be seen as the initial condition or starting value before any changes (dictated by the slope) occur. Recognizing the y-intercept helps in understanding the behavior of a line and the context of a situation it might represent.

Slope

The slope of a line in a linear equation tells you how steep the line is. In the equation \( y = mx + b \), the slope is represented by 'm'. It is calculated as the "rise over run," or the change in the y-values divided by the change in the x-values as you move along the line.

• For example, a slope of \( \frac{1}{2} \) means that for every 2 units you move horizontally, the line rises by 1 unit.
• A positive slope, like \( \frac{1}{2} \), indicates that the line increases from left to right on the graph.
• A negative slope means the line decreases from left to right.

Understanding the slope is crucial because it dictates the direction and angle of the line on a graph. It's a measure of the rate of change between the variables in the equation. In practical terms, the slope can signify speed, direction, or any number of other factors, depending on what the "x" and "y" represent in a real-world scenario.