Problem 6 - Thinking Mathematically

Question

Use the \(x\) - and \(y\)-intercepts to graph each linear equation. \(x+3 y=6\)

Step-by-Step Solution

Answer

The y-intercept is at the point (0,2), and the x-intercept is at the point (6,0). So, these points can be plotted and a line can be drawn passing through these points to graph the given linear equation.

Step 1: Find the y-intercept

To find the y-intercept, set x equal to 0 and solve the given equation for y: \n Substitute \(x = 0\) into the equation \(x+3 y = 6\) to get \n \(0+3y = 6\), \n which simplifies to \n \(3y = 6\). \n Divide both sides of the equation by 3 to isolate y: \n \(y = 6/3 = 2\). \n Thus, the y-intercept is 2.

Step 2: Find the x-intercept

To find the x-intercept, set y equal to 0 and solve the equation for x: \n Substitute \(y = 0\) into the equation \(x + 3y = 6\) to get \n \(x + 3*0 = 6\), \n which simplifies to \n \(x = 6\). \n Thus, the x-intercept is 6.

Step 3: Plot the graph

Now, these intercepts can be plotted on a graph: \n The y-intercept (when x=0) is at the point (0,2), and the x-intercept (when y=0) is at the point (6,0).\n These two points can now be plotted on the graph. Draw a straight line through these points to represent the equation.

Key Concepts

x-intercept

The x-intercept of a linear equation is the point where the graph of the equation crosses the x-axis. At this point, the value of y is always zero. Finding the x-intercept is straightforward:

• Set the y-value to zero in the equation.
• Solve the resulting equation to find the value of x.

In the equation given, which is \(x + 3y = 6\), setting \(yequals0\) simplifies it to \(x = 6\). This means the x-intercept is at the coordinate (6, 0).This point tells us that, when y is zero, x equals 6. It's an essential point for graphing as it helps define the line's path on the graph. Remember, the x-intercept emphasizes the behavior of the equation when the output or "height" (y) is zero.

y-intercept

The y-intercept represents the point where the graph crosses the y-axis. At this intercept, the x-value is always zero. Finding the y-intercept is a similar process to finding the x-intercept:

• Set the x-value to zero in the equation.
• Solve the resulting equation to find the value of y.

Using the given equation \(x + 3y = 6\), when you substitute \(x = 0\), it simplifies to \(3y = 6\). Solving for y leads to \(y = 2\), showing the y-intercept at the coordinate (0, 2).This point signifies that when x is zero, y equals 2. The y-intercept is crucial for understanding where the line will cross the y-axis on the graph. It also provides a clear point that can be used to begin graphing the equation, indicating the starting point on the y-axis.

graphing linear equations

Graphing linear equations involves plotting the line that represents all possible solutions to the equation on a coordinate plane. It begins with finding and plotting the intercepts—both x- and y-intercepts are pivotal.To graph the equation \(x + 3y = 6\), follow these steps:

• First, find the x-intercept, which is the point (6, 0).
• Then, find the y-intercept, which is the point (0, 2).
• Mark these points on a graph.
• Draw a line through them to complete the graph of the equation.

This line represents all combinations of x and y that satisfy the equation. A graph not only visually shows solutions but also helps in understanding relationships between variables. With the intercepts known, setting accurate scales on both axes ensures a precise depiction of the line. Remember, this method provides a clear visual path from intercept to intercept, allowing for easy comprehension of linear equations.