Problem 4 - Algebra 1: Concepts and Skills

Question

find the slope and y-intercept of the equation. $$y=8.5 x$$

Step-by-Step Solution

Answer

The slope of the line is \(8.5\) and the y-intercept is \(0\).

Step 1: Identify the Form of the Equation

Determine that the equation is in slope-intercept form (\(y=mx+b\)). The equation given is \(y=8.5x\), which matches the required form.

Step 2: Identify the Slope

The coefficient of \(x\) is the slope (\(m\)). In this case, the coefficient of \(x\) is \(8.5\), hence the slope of this line is \(m=8.5\).

Step 3: Identify the Y-intercept

The y-intercept (\(b\)) is the value of \(y\) when \(x=0\). In this equation, there is no constant term, that means the line passes through the origin. Hence, the y-intercept is \(b=0\).

Key Concepts

Calculating Slope

Understanding how to calculate the slope of a line is a fundamental skill in algebra which allows one to understand the rate of change between two variables. The slope is represented by the letter 'm' and is calculated by the rise over run, which means the change in 'y' over the change in 'x'.

In the equation provided, we see it in the form of \(y = 8.5x\), where the coefficient of \(x\) represents the slope. There are no complications here because the slope is clearly stated as 8.5. This slope tells us that for every unit of increase in \(x\), \(y\) increases by 8.5 units. It's essential to notice that if this coefficient is positive, the line slopes upwards as it moves from left to right, and if it were negative, it would slope downwards.

Identifying Y-Intercept

The y-intercept is the point where the line crosses the y-axis. It's the value of \(y\) when \(x\) is zero and is represented as the 'b' in the slope-intercept equation \(y = mx + b\).

In the provided exercise, our equation is \(y=8.5x\) which lacks the \(b\) term. This indicates that the y-intercept is \(0\), meaning the line crosses the origin at point \(0,0\). Identifying the y-intercept is crucial for graphing the line and understanding its equation. If there were a number added to the end of the equation (for example, \(y=8.5x+2\)), that number would be our y-intercept and the point where the line crosses the y-axis would be \(0,2\).

Linear Equations

Linear equations are the simplest form of equations used to describe a straight line. They follow the standard slope-intercept form of \(y=mx+b\), where 'm' represents the slope and 'b' denotes the y-intercept. A linear equation allows us to see the relationship between two variables and can be easily graphed on a coordinate plane.

To graph a linear equation, start by plotting the y-intercept on the y-axis. From this point, use the slope to determine the next point. Ensure that you move positively or negatively along the x-axis depending on the sign of the slope, then up or down to account for the slope value. For instance, with a slope of 8.5, you would move 1 unit to the right and 8.5 units up from the y-intercept. Repeat this step to find another point, then draw a line through these points, and you have graphed your linear equation.