Problem 2 - Elementary Algebra

Question

The following equation are in slope-intercept form. In each case, specify the slope and \(y\) -intercept. $$ y=-4 x+2 ; \quad m=\quad b= $$

Step-by-Step Solution

Answer

Answer: The slope (m) of the equation is $$-4$$, and the y-intercept (b) is $$2$$.

Step 1: Identify the given equation's form

The given equation is $$y = -4x + 2$$. Observe that this equation is already in slope-intercept form, where the coefficients of the x-term and constant term represent the slope and y-intercept respectively.

Step 2: Determine the slope (m)

In the given equation, the coefficient of x is -4. This is the slope (m), hence $$m=-4$$.

Step 3: Determine the y-intercept (b)

In the given equation, the constant term is 2. This is the y-intercept (b), hence $$b=2$$.

Step 4: Write down the final answer

In the equation $$y = -4x + 2$$, the slope (m) is $$-4$$, and the y-intercept (b) is $$2$$.

Key Concepts

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols; it is a language of its own that allows us to formulate and solve problems in a systematic way. It involves everything from solving simple equations to studying abstractions such as groups, rings, and fields.

When we work with algebraic expressions and equations, we are often looking for unknown values that make the equation true. In the context of the exercise, we're using algebra to describe straight lines on a graph using equations. Algebra is the foundation for understanding linear equations, which are expressions that depict a line on a coordinate plane.

Linear Equations

Linear equations are algebraic expressions that represent a straight line when graphed on a coordinate plane. They have one or more variables with no exponents or powers higher than one and appear in the form of \[\begin{equation} ax + by = c \text{or} \[\begin{equation} y = mx + b \text{(slope-intercept form)}.\end{equation}\] \end{equation}\] Here, 'a', 'b', and 'c' are constants with 'a' and 'b' not both zero, while 'm' represents the slope and 'b' is the y-intercept in the slope-intercept form.

Understanding how to read and graph these equations is fundamental in algebra. The slope-intercept form particularly tells us the steepness of a line and where it crosses the y-axis, allowing quick sketching and analysis of the line's behavior.

Slope

What is Slope?
The slope of a line, often denoted by 'm', measures the steepness, incline, or declination of the line. It is calculated as the change in y (vertical change) over the change in x (horizontal change), commonly expressed as 'rise over run'.

For the equation \[\begin{equation} y = -4x + 2 \text{,}\end{equation}\] the slope is -4. It tells us that for every one unit we move to the right along the x-axis, the line falls four units down because of the negative sign. A positive slope means the line rises, and a larger absolute value means the line is steeper.

Y-intercept

Understanding the Y-intercept
The y-intercept of a line is the point where the line crosses the y-axis. It's represented in the slope-intercept form of a linear equation as 'b'. Specifically, in the equation \[\begin{equation} y = -4x + 2 \text{,}\end{equation}\] the y-intercept is 2.

This means that if you start at the origin (0,0) and follow the line, it would intersect the vertical y-axis at the point (0, 2). The y-intercept is a key element in graphing as it provides a starting point for drawing the line on a graph. When you know both slope and y-intercept, you can fully describe any line on a plane, and solve various applied problems.