Problem 44 - Algebra 1

Question

Write an equation of the line in slope-intercept form. The slope is \(2 ;\) the \(y\) -intercept is \(-5\)

Step-by-Step Solution

Answer

The equation of the line in slope-intercept form is \(y = 2x - 5\).

Step 1: Identify and assign the slope and y-intercept

The problem provides us with a slope (\(m\)) of \(2\) and a y-intercept (\(c\)) of \(-5\).

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

Plugging these values into the slope-intercept form equation \(y = mx + c\) results in \(y = 2x - 5\).

Key Concepts

Writing Equations of Lines

Writing equations of lines in slope-intercept form is a foundational skill in algebra that enables you to represent the relationship between two variables in a linear fashion. The slope-intercept form of a line is given by the equation \(y = mx + b\), where \(m\) is the slope of the line, and \(b\) is the y-intercept. The slope represents the rate at which the dependent variable (usually \(y\)) changes with respect to the independent variable (usually \(x\)), while the y-intercept represents the value of \(y\) when \(x = 0\).

To write an equation, you first need to know these two pieces of information. If the problem provides you with the slope and y-intercept directly, like in our exercise example, the process is straightforward. But sometimes, you might have to calculate the slope from two given points using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\), or the y-intercept by setting \(x\) to zero and solving for \(y\). Knowing how to manipulate these variables and translate them into an equation is a key skill in creating mathematical models for various real-world applications.

Identifying Slope and Y-Intercept

Being able to identify the slope and y-intercept from a given equation or data is crucial in understanding linear relationships. The slope, typically represented as \(m\), signifies the direction and steepness of the line. A positive slope means the line inclines upward as it moves from left to right while a negative slope means the line declines. If the slope is zero, the line is horizontal, implying no change in \(y\) as \(x\) changes; a line with an undefined slope is vertical because there is no change in \(x\) as \(y\) changes.

The y-intercept, represented as \(b\), is the point where the line crosses the y-axis. To identify it from an equation, look for the constant term when the equation is in slope-intercept form. In other words, it's the value of \(y\) when \(x = 0\). This point is also where the graph of the equation will start if you were to plot it on a coordinate plane. Understanding these elements helps in sketching the graph of the line and predicting the behavior of the linear relationship described by the equation.

Substituting Values in Slope-Intercept Form

Once you have the slope-intercept form of a linear equation (\(y = mx + b\)), you can easily substitute values for \(x\) to find corresponding \(y\) values, which is helpful when plotting the line or when you need to make predictions based on the model. For example, if an equation of a line is \(y = 3x - 4\), and you want to find the \(y\) when \(x = 2\), you would substitute \(2\) for \(x\) and compute \(y = 3(2) - 4\), which simplifies to \(y = 2\).

This substitution process is essential not just in plotting but also in solving systems of linear equations, where you set two linear equations equal to each other and solve for the variables. When given a slope and a y-intercept, as you saw in the provided exercise, substituting these values into the slope-intercept equation allows you to form the complete equation that represents the line. This can then be a tool for further analysis, such as understanding how changes in one variable affect another within a linear relationship.