Problem 27 - Algebra 1: Concepts and Skills

Question

Graph the equation. $$y=2 x-1$$

Step-by-Step Solution

Answer

The graph of the equation \(y=2x-1\) is a straight line that crosses the y-axis at -1 and rises with a slope of 2.

Step 1: Identify the slope

In the given equation \(y = 2x -1\), the coefficient of x is the slope of the line. Here, the slope is 2.

Step 2: Identify the y-intercept

The y-intercept is the value of y when x equals 0. In the equation, the y-intercept is -1, which is the constant term.

Step 3: Plot the y-intercept

Begin graphing by marking the y-intercept (-1) on the y axis on the Cartesian plane.

Step 4: Plot the slope

From the y-intercept, use the slope to find the next point. The slope is 2, which means for every step to the right on the x axis, move up by 2 steps on the y axis. This gives another point through which the line passes.

Step 5: Draw the line

Draw a straight line through the two points plotted. This line represents the solution set for the given equation.

Key Concepts

Slope

The slope is a measure that shows how steep a line is on the graph. It's the change in the y-value for a corresponding change in the x-value. This is often referred to as "rise over run." In the equation of the form \(y = mx + b\), the slope is represented by \(m\). For example, in the equation \(y = 2x - 1\), the slope \(m\) is 2.

This tells us two important things:

• The line moves up by 2 units on the y-axis for every 1 unit it moves to the right on the x-axis.
• A positive slope, as in this example, means the line goes up from left to right. Conversely, a negative slope would mean the line goes down from left to right.

Understanding the slope is crucial because it gives us a quick way to sketch out a line just by knowing this one number. It essentially defines how the line is oriented on the graph.

Y-Intercept

The y-intercept is where the graph of the equation crosses the y-axis. In the equation \(y = mx + b\), the y-intercept is represented by \(b\). This point occurs where \(x = 0\), so you can quickly identify it in linear equations by looking at the constant term.

For instance, in the equation \(y = 2x - 1\), the y-intercept is -1. This means:

• When you plug in \(x = 0\), \(y = -1\).
• On the graph, the line will cross the y-axis at the point (0, -1).

Knowing the y-intercept helps you start the process of graphing a linear equation since it's your initial point on the Cartesian plane. From this point, you can further apply the slope to draw the line.

Cartesian Plane

The Cartesian plane is the two-dimensional plane we use for graphing equations like \(y = mx + b\). It consists of an x-axis, which is horizontal, and a y-axis, which is vertical. These axes intersect at the origin, point (0, 0).

Grasping the layout of the Cartesian plane is essential for plotting any equation:

• The area to the right of the y-axis is the positive x region, and to the left is the negative x region.
• Similarly, above the x-axis is the positive y region, and below is the negative y region.

When plotting the equation \(y = 2x - 1\), you'd mark starting points based on the y-intercept and then use the slope to determine other points. Understanding the quadrants and how to navigate the plane makes graphing equations a systematic process.

Linear Equations

Linear equations are equations of the first degree, which means their highest power of the variable(s) is one. They have the general form \(y = mx + b\), and on a graph, a linear equation corresponds to a straight line.

Key characteristics of linear equations include:

• They graph as straight lines, thus the term "linear."
• They have constant slopes, meaning the steepness or inclination doesn't change over the line.
• The equation \(y = 2x - 1\) is a linear equation, representing a line that intersects the y-axis at -1 and has a slope of 2.

Understanding linear equations is crucial for solving and graphing mathematics problems related to real-world situations, like predicting trends or finding intersections.