Question

Find the slope and the y-intercept of the line. $$y=3 x-6.5$$

Short answer

The slope of the line is 3, and the y-intercept is -6.5

Step-by-step solution

Identify the slope

The coefficient of x in the equation is the slope. Looking at the equation, we can therefore say the slope of the line, \(m\), is 3

Identify the y-intercept

The constant term is the y-intercept in the slope-intercept form. In our equation, the y-intercept, \(c\), is -6.5

Key concepts

Slope-Intercept Form

When dealing with linear equations, the slope-intercept form is a popular and convenient way to write them. It is expressed as
\[ y = mx + b \]
where \( m \) represents the slope and \( b \) indicates the y-intercept of the line. This form allows you to quickly understand how the line behaves. The slope, \( m \), tells us how steep the line is - whether it goes uphill or downhill as you move from left to right across the graph. If the slope is positive, the line rises, and if it is negative, the line falls. The y-intercept, \( b \), tells us the point where the line crosses the y-axis. This is where the graph of the equation will touch or cross the vertical y-axis on a graph.

Coefficient of x

In linear equations written in slope-intercept form, the coefficient of x, usually denoted as \( m \), is the multiplier of the \( x \) variable. It is key in defining the angle of the line with respect to the x-axis. The larger the coefficient, the steeper the line.

Real-World Application

Understanding the coefficient is essential in various practical applications, such as calculating grade percentages in education or understanding the rate of change in economics. In the case of \( y = 3x - 6.5 \) from the exercise, the coefficient of x is 3, meaning for every one unit that x increases, y increases by three units.

Constant Term

The constant term in an equation is the value that does not change; it is the number without a variable attached to it.

Function of the Constant Term

In the slope-intercept form of a linear equation, the constant term represents the y-intercept which determines where the line will cross the y-axis. It is the starting value of y when x is zero. So, for our example equation \( y = 3x - 6.5 \), the constant term is -6.5. This indicates that the line will cross the y-axis below the origin at the point (0, -6.5).

Linear Equations

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. Linear equations produce a straight line when graphed on a coordinate plane.

Characteristics of Linear Equations

Linear equations have a constant rate of change, which is represented by their slope. The standard form is \( Ax + By = C \), but they are often converted into slope-intercept form, \( y = mx + b \), for simpler interpretation of the slope and y-intercept. They are fundamental in algebra and are widely used in fields like physics, economics, and engineering to describe relationships where one variable is a constant multiple of another, modified by a constant.