Problem 77 - Algebra 1: Concepts and Skills

Question

Find the slope and \(y\) -intercept of the graph of the equation. (Lesson 4.7) $$ y-8 x=2 $$

Step-by-Step Solution

Answer

The slope of the given equation is 8 and the y-intercept is 2.

Step 1: Rearrange the Equation

Rearrange the given equation into slope-intercept form. This can be done by isolating y on one side of the equation: \(y = 8x + 2\)

Step 2: Identify the Slope and Y-intercept

Now that the equation is in the slope-intercept form, the slope \(m\) corresponds to the coefficient of \(x\), and the \(y\)-intercept \(b\) corresponds to the constant.

Key Concepts

Finding Slope

The slope of a line is a measure of its steepness and direction on a coordinate plane. It's represented by the letter 'm' in linear equations. When an equation is in the form of \(y = mx + b\), finding the slope is straightforward; it's simply the coefficient of the variable \(x\).

To find the slope from the equation \(y - 8x = 2\), we rearrange the equation to get \(y = 8x + 2\). Here, the number 8, which is multiplying the variable \(x\), is the slope. This indicates that for each unit increase in \(x\), \(y\) increases by 8 units, making the line rise sharply and to the right if you're looking at a graph. If the slope were negative, it would imply a decrease in \(y\) with an increase in \(x\), sloping the line down and to the right.

Y-Intercept

The y-intercept is the point where the line crosses the y-axis on a graph. Represented by the letter 'b' in the slope-intercept form of a line (\(y = mx + b\)), the y-intercept is a constant value that signifies the value of \(y\) when \(x\) is zero.

In our example \(y = 8x + 2\), the y-intercept is 2. This means that when \(x\) is 0, the value of \(y\) will be 2, placing the y-intercept at point (0, 2) on the graph. Knowing the y-intercept is crucial because it gives us a starting point for graphing the linear equation and understanding the line's vertical position relative to the origin.

Linear Equations

Linear equations are algebraic equations where each term is either a constant or the product of a constant and the first power of a variable. They graph as straight lines, and the most common form to express them is the slope-intercept form, which is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

Linear equations represent relationships between two variables, often \(x\) and \(y\), where each value of \(x\) corresponds to exactly one value of \(y\), resulting in a set of points that create a straight line when plotted on a coordinate graph. They are fundamental in expressing various real-life situations, such as motion at a constant speed, and in the fields of economics, sciences, and engineering, to describe various linear relationships.