Problem 6 - Algebra 1: Concepts and Skills

Question

find the slope and y-intercept of the equation. $$y-x=3$$

Step-by-Step Solution

Answer

The slope of the given equation is 1 and the y-intercept is 3.

Step 1: Rearrange the Equation

To identify the slope and the y-intercept, rearrange the equation \(y - x = 3\) into the form \(y = mx + b\), which will then read as \(y = x + 3\).

Step 2: Identify the Slope

Looking at the rearranged equation \(y = x + 3\), the coefficient of \(x\), which is 1, is recognized as the slope \(m\). Therefore, the slope of the equation is 1.

Step 3: Identify the y-intercept

The y-intercept is represented by \(b\) in the equation \(y = mx + b\). Looking at the equation \(y = x + 3\), the y-intercept, \(b\), is 3.

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form is essential for analyzing linear equations. This form is represented as \(y = mx + b\) , wherein \(m\) stands for the slope and \(b\) indicates the y-intercept. The slope measures the steepness of the line and its direction. If the slope is positive, the line ascends from left to right; if negative, it descends. The slope of 1, as in the given equation \(y = x + 3\), signifies a line that ascends at a 45-degree angle.

• The y-intercept \(b\) is the point where the line crosses the y-axis. This point corresponds to the value of \(y\) when \(x = 0\).
• In the exercise, the y-intercept is 3, which means the line intersects the y-axis at point (0, 3).
• Remember, the slope-intercept form makes graphing more straightforward, as you can plot the y-intercept first and then use the slope to find other points.

Linear Equations

Linear equations form the foundation of algebra and represent straight lines on a graph. Their general form is \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants. To graph a linear equation effectively, you often first convert it to the slope-intercept form. The step-by-step solution provided, transforms \(y - x = 3\) to slope-intercept form to expose the slope and y-intercept.

Here's why understanding the conversion to slope-intercept form is important:

• It simplifies finding the slope and y-intercept – the two critical features of a linear graph.
• The equation in slope-intercept form allows for quick visualization of the line's behavior on the coordinate plane.
• Recognizing \(y = mx + b\) as the standard form for linear equations in slope-intercept form supports the categorization of equations and comparison of their lines' characteristics such as parallelism or intersection.

Graphing Linear Equations

Graphing linear equations is a graphical way of representing solutions to equations. It is not only a visual aid but also a powerful tool to understand the properties of the linear equation.

When given an equation like \(y = x + 3\), here's how you graph it:

• Start at the y-intercept (0, 3) on the y-axis. This is where the line will cross the y-axis.
• From the y-intercept, use the slope as a ratio of \(rise/run\). In this case, a slope of 1 means you move up one unit and right one unit on the graph to find another point.
• Mark the second point and draw a line through the two points, extending it in both directions. This line represents all the solutions to the equation.
• Don't forget to label your axes and scale your graph appropriately for a more accurate representation.

Graphing an equation allows you to see at a glance the set of points that make the equation true. This can be particularly helpful for visual learners and when solving problems involving intersection and regions defined by linear inequalities.