Problem 28 - Introductory and Intermediate Algebra for College Students 4th

Question

Use intercepts and a checkpoint to graph each equation. $$2 x-y=5$$

Step-by-Step Solution

Answer

The x-intercept is 2.5, the y-intercept is -5 and a check point is (1,-3). The line passing through these points is the graph of the equation.

Step 1: Find the X-intercept

To find the x-intercept, simply set \(y=0\) into the equation and solve for \(x\). So we have \(2x - 0 = 5\), thus, \(x=5/2=2.5\). So the x-intercept is 2.5.

Step 2: Find the Y-intercept

Now find the y-intercept by setting \(x=0\) in the equation and solving for \(y\). This leads to \(2*0 - y = 5\), which simplifies to \(-y = 5\). Therefore, \(y=-5\). So the y-intercept is -5.

Step 3: Choose a Checkpoint

We choose another point other than the intercepts, to ensure the precision of our line. Let's select \(x=1\), this gives \(2*1 - y = 5\). So \(y=-3\). Hence, the checkpoint is (1,-3).

Step 4: Graph the Equation

Plot the x-intercept (2.5, 0), the y-intercept (0, -5) and the checkpoint (1, -3) on the graph. Draw a line passing through these points. This line represents the graph of our equation.

Key Concepts

Understanding the X-Intercept

The x-intercept is the point where a graph crosses the x-axis, which means the value of y is zero at this point. To find the x-intercept from an equation like \(2x - y = 5\), you substitute \(y = 0\) into the equation and solve for \(x\).

In our example, setting \(y = 0\) gives us \(2x = 5\), which means \(x = 2.5\). Therefore, the x-intercept of this equation is \((2.5, 0)\). This is a crucial step, as it provides one of the reference points necessary to sketch the graph of a linear equation.

Finding the Y-Intercept

The y-intercept is the point where the graph crosses the y-axis. At this intersection, the value of \(x\) is zero. To determine the y-intercept from the equation \(2x - y = 5\), set \(x = 0\) and solve for \(y\).

Doing this in our example leads to \(-y = 5\), which means that \(y = -5\). Hence, the y-intercept is \((0, -5)\). Knowing the y-intercept helps to anchor another point with which to draw the line representing our equation on a graph.

Navigating the Coordinate Plane

A coordinate plane is a two-dimensional space defined by a horizontal axis (x-axis) and a vertical axis (y-axis). It is used to plot points, lines, and curves. In the context of graphing a linear equation like \(2x - y = 5\), you plot points such as the x-intercept and y-intercept on this plane.

Every point on the plane corresponds to an x-coordinate and a y-coordinate, which are written as an ordered pair \((x, y)\). By identifying intercepts, we can draw a line through these points to represent our equation visually. This representation allows us to understand the relationship between x and y values within the equation.

Exploring the Slope-Intercept Form

The slope-intercept form of a linear equation is expressed as \(y = mx + b\), where \(m\) is the slope of the line and \(b\) is the y-intercept. The slope indicates how steep the line is, showing the change in y for every unit change in x. To convert our equation \(2x - y = 5\) to this form, solve for \(y\).

This yields \(y = 2x - 5\), which shows the slope \(m = 2\) and the y-intercept \(b = -5\). Understanding the slope-intercept form allows for easier graphing and interpretation of the line's properties, such as direction and rate of change.