Problem 44 - Algebra 2

Question

Find the slope and \(y\) -intercept of the graph of each function. $$ y=5 x $$

Step-by-Step Solution

Answer

The slope of the graph of the function is 5 and the y-intercept is 0.

Step 1: Identify the Slope

The standard form for the equation of a line is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. In the given equation \(y = 5x\), the coefficient of \(x\) is 5, which represents the slope of the line.

Step 2: Identify the Y-intercept

Since there is no constant added or subtracted from \(5x\) in the equation \(y = 5x\), it implies that the y-intercept, \(b\), is 0. Therefore, the line crosses the y-axis at the origin (0,0).

Key Concepts

Linear Equations

Understanding linear equations is crucial in grasping the basics of algebra. Simply put, a linear equation represents a straight line when plotted on a graph. It takes the form of y = mx + b, where y represents the dependent variable, x is the independent variable, m is the slope of the line, and b is the y-intercept—or where the line crosses the y-axis.

Linear equations are straightforward because they describe a constant rate of change. For instance, if we look at an equation like y = 5x, it tells us that for every unit increase in x, the value of y increases by five units. This idea of a 'constant rate of change' is what makes the line straight, and linear equations can be easily recognized by their degree—always 1 for both x and y.

Slope of a Line

The slope is a measure of how steep a line is. Formalized, it is the rate at which the y value of a line changes with respect to the x value. In the equation y = mx + b, m denotes the slope. It's calculated by determining the 'rise' over the 'run' between two distinct points on the line.

If a line rises 3 units for every 4 units it runs horizontally, the slope would be 3/4. If the line falls as it moves to the right, the slope is negative. In the exercise y = 5x, the slope is 5, which means for every unit that x increases, y increases by 5 units—indicating a rather steep line. This also implies the line rises 5 units for every 1 unit of run.

Y-Intercept

The y-intercept can be described simply as the spot where the line crosses the y-axis. This happens when the value of x is zero. In the equation format y = mx + b, b gives us this value directly.

In practical terms, it’s often the starting value or initial condition in real-world situations—for example, the initial amount in a bank account before interest is applied. In the case of our exercise y = 5x, since no other number is being added to 5x, our y-intercept is 0. This means our line crosses the y-axis right at the origin, the point (0,0), showing no initial value before the increase dictated by our slope.

Standard Form of a Line

The standard form of a line's equation is often expressed as Ax + By = C. It differs slightly from the slope-intercept form in the way it's structured. A, B, and C are integers, and the coefficients A and B are not both zero.

One benefit of the standard form is it's useful when dealing with vertical and horizontal lines, which can be problematic in slope-intercept form since the slope would be undefined or zero, respectively. To convert the slope-intercept equation y = 5x to standard form, you might rewrite it as -5x + y = 0, ensuring that the coefficients meet the integer requirement.