Problem 16 - Algebra 1: Concepts and Skills

Question

Write in slope-intercept form the equation of the line described below. $$ m=10, b=0 $$

Step-by-Step Solution

Answer

The line's equation in slope-intercept form which has a slope of 10 and y-intercept of 0 is \( y = 10x \).

Step 1: Identify the Given Slope and Intercept

A line's equation is given in the form \( y = mx + b \), where \( m \) represent the slope of the line and \( b \) is the y-intercept. From the exercise, the slope \( m \) provided is 10, and the y-intercept \( b \) is 0.

Step 2: Substituting the Values into the Equation

Replace \( m \) with 10 and \( b \) with 0 in the line's equation. The equation becomes: \( y = 10x + 0 \) .

Step 3: Simplify the Equation

The final equation of the line will be \( y = 10x \) because adding 0 to anything does not change its value.

Key Concepts

Linear Equations

When we talk about linear equations, we're discussing a fundamental component of algebra. These equations represent straight lines when graphed on a coordinate plane and are commonly expressed in the form of y = mx + b, known as the slope-intercept form. The aspect that classifies them as 'linear' is that the variables are to the first power, meaning there are no squares, cubes, or higher powers in the equation.

In our exercise, we're asked to write the equation of a line with a given slope and y-intercept in this specific form. This structure is particularly useful because it provides a straightforward representation of the line, showcasing how for every unit change in x (the independent variable), y (the dependent variable) will change proportionally, based on the slope's value.

Slope of a Line

Understanding the slope of a line is crucial when working with linear equations. The slope, indicated by the variable m in the slope-intercept form equation, is a measure of the steepness or incline of the line. It's calculated as the 'rise over run,' representing the change in y (vertical change) over the change in x (horizontal change).

A positive slope means the line is rising as it moves from left to right, while a negative slope indicates the line is falling. A slope of zero suggests a horizontal line, and an undefined slope corresponds to a vertical one. In our exercise example, the slope is 10, meaning for every single unit increase in x, the value of y increases by 10 units.

Y-intercept

The y-intercept of a line is the point where the line crosses the y-axis. This is another key element of the slope-intercept form, denoted as b in the equation y = mx + b. The y-intercept represents the value of y when x is zero, providing a fixed point from which the line can be graphed.

In the context of our exercise, the y-intercept is given as 0. This tells us that the line crosses the origin of the coordinate plane, which is the point (0,0). Therefore, even after applying the slope, as demonstrated in the step-by-step solution, the line will not be shifted up or down on the graph.