Problem 21 - Introductory Algebra for College Students

Question

Use intercepts and a checkpoint to graph equation. \(x+3 y=6\)

Step-by-Step Solution

Answer

The x-intercept is (6,0), the y-intercept is (0,2), and the checkpoint is (1,5/3). These are the points used to graph the equation \(x+3y=6\).

Step 1: Find the x-intercept

To find the x-intercept, let \(y = 0\) in the given equation \(x+3y=6\). So, \(x + 3*0 = 6\), making the x-intercept (6,0).

Step 2: Find the y-intercept

To find the y-intercept, let \(x = 0\) in the given equation \(x+3y=6\). So, \(0+3y = 6\), after simplifying we get \(y = 2\), making the y-intercept (0,2).

Step 3: Find a Checkpoint

To find a checkpoint, pick a convenient value for either x or y (excluding already used zeros) in the equation \(x+3y=6\), and solve for other variable. Lets pick \(x = 1\), then the equation becomes \(1+3y=6\), solving for y we get \(y = \frac{5}{3}\). So, the checkpoint is (1, 5/3).

Step 4: Graph the Equation

Plot the x-intercept (6,0), the y-intercept (0,2), and the checkpoint (1,5/3) on a graph. Connect these points with a straight line to graph the equation.

Key Concepts

Understanding the X-Intercept

The x-intercept is a crucial point that helps us graph a linear equation. It is the point where the line crosses the x-axis. To find the x-intercept, you set the value of y to zero in the equation and solve for x. This is because any point on the x-axis has a y-coordinate of zero.
In the linear equation provided, which is \( x + 3y = 6 \), you substitute \( y = 0 \) to find the x-intercept. That changes the equation to \( x + 3*0 = 6 \), simplifying to \( x = 6 \). Thus, the x-intercept is the point \( (6,0) \).
Knowing the x-intercept gives you a solid starting point on the horizontal axis for your graph.

Understanding the Y-Intercept

The y-intercept is the point where the line crosses the y-axis. To find it, set x to zero and solve for y because all points on the y-axis have an x-coordinate of zero. This point is vital because it tells us where the line will encounter the y-axis.
In the equation \( x + 3y = 6 \), setting \( x = 0 \) means it simplifies to \( 3y = 6 \). Solving for y gives \( y = 2 \), meaning the y-intercept is \( (0,2) \).
These intercepts are essential because they provide two fixed points through which you can draw the line that represents your equation on a graph.

Utilizing Checkpoints for Accuracy

Sometimes just intercepts are not enough to sketch a precise line, so we use additional points known as checkpoints. These points help ensure the line is accurately drawn, especially if it doesn't seem straight.
To find a checkpoint, choose a value for either x or y that isn’t zero (since 0 has been used to find intercepts) and solve the equation again. For the equation \( x + 3y = 6 \), let's try \( x = 1 \). This turns the equation into \( 1 + 3y = 6 \). Solving for y gives \( y = \frac{5}{3} \). Thus, a checkpoint is \( (1, \frac{5}{3}) \).
Using intercepts and checkpoints together ensures your graphed line accurately represents the equation, confirming the slope and alignment of the line.