Question

Write an equation in slope-intercept form of the line with the given slope that passes through the given point. $$m=-3,(0,4)$$

Short answer

The equation is \( y = -3x + 4 \).

Step-by-step solution

Identify Components

First, identify the components provided in the problem. We have the slope of the line, denoted by \( m = -3 \), and the point through which the line passes, which is \((0, 4)\).

Recall the Slope-Intercept Form

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

Determine the Y-Intercept

Since the point \((0, 4)\) is on the y-axis, it gives us the y-intercept directly. Thus, the y-intercept \( b = 4 \).

Write the Equation

Insert the values of the slope \( m = -3 \) and y-intercept \( b = 4 \) into the slope-intercept form. The equation of the line becomes \( y = -3x + 4 \).

Key concepts

Linear Equations

Linear equations are fundamental mathematical tools used to represent straight lines in a coordinate plane. They are expressed in the form of an equation, where each point on the line corresponds to a solution to the equation. In its most straightforward form, a linear equation is represented as \( y = mx + b \). Here, \( m \) is the slope of the line, and \( b \) is the y-intercept. This form is known as the slope-intercept form. Linear equations are essential for:

• Determining the relationship between two variables.
• Solving problems in geometry and algebra.
• Modeling real-world situations like predicting trends and analyzing data.

Understanding the basic components of linear equations can simplify many mathematical problems and aid in visualizing how changes in variables affect outcomes.

Slope of a Line

The slope of a line, represented by \( m \), is a key component in understanding linear equations. It indicates how steep or flat a line is on a graph. Mathematically, the slope is calculated as the change in the vertical direction (\( \Delta y \)) over the change in the horizontal direction (\( \Delta x \)). In formulaic terms, this is expressed as \( m = \frac{\Delta y}{\Delta x} \).

• A positive slope means the line goes upward as you move from left to right.
• A negative slope indicates the line goes downward.
• A zero slope represents a horizontal line.
• An undefined slope means the line is vertical.

The slope provides valuable information about the direction and inclination of the line, helping to predict and understand relationships between variables.

Y-Intercept

The y-intercept of a line is the point where the line crosses the y-axis. In the equation of a line in slope-intercept form, \( y = mx + b \), \( b \) represents the y-intercept. This can be determined directly if a point provided is on the y-axis, such as the point \((0, 4)\) in our example. Here, the y-intercept \( b \) is 4.Key points about the y-intercept:

• It shows where the line intersects the y-axis.
• It is important for estimating the starting value of a function when \( x = 0 \).
• In real-world scenarios, it can represent a fixed value like an initial cost or starting condition.

Understanding the y-intercept is crucial in graphing lines and interpreting linear relationships in both abstract and applied contexts.