Problem 20 - Introductory and Intermediate Algebra for College Students 4th

Question

Use intercepts and a checkpoint to graph each equation. $$x+y=6$$

Step-by-Step Solution

Answer

The graph of the equation \( x + y = 6 \) would intercept the x-axis at the point (6,0), the y-axis at point (0,6) and it would also pass through the checkpoint (3,3).

Step 1: Identifying the x-intercept

To find the x-intercept, set y equal to zero and solve for x. That's because the x-intercept is the point where the graph crosses the x-axis, which is when \( y = 0 \). Hence, if \( y = 0 \), then the equation \( x + 0 = 6 \) simplifies to \( x = 6 \). Therefore, the x-intercept is at the point (6,0).

Step 2: Identifying the y-intercept

To find the y-intercept, set x equal to zero and solve for y. This is because the y-intercept is the point where the graph crosses the y-axis, which is when \( x = 0 \). Hence, If \( x = 0 \), then the equation \( 0 + y = 6 \) simplifies to \( y = 6 \). Therefore, the y-intercept is at the point (0, 6).

Step 3: Selecting a suitable checkpoint

A checkpoint is a point on the grid that is used to check the accuracy of the graph. A good checkpoint for this equation would be the point (3, 3) because if you substitute these values into the equation, it holds true: \( 3 + 3 = 6 \).

Step 4: Drawing the graph

Plot the x and y intercepts and the checkpoint on the graph paper. Then, draw a straight line through these points. The line represents all the possible solutions to the equation.

Key Concepts

Intercepts

Intercepts are crucial in graphing linear equations. They help in quickly sketching the graph by providing key points where the line crosses the axes. In our example, the equation is given as \(x + y = 6\).

• The x-intercept indicates where the line crosses the x-axis and is found by setting \(y = 0\).
• The y-intercept indicates where the line crosses the y-axis and is found by setting \(x = 0\).

By identifying these intercepts, you have a good basis for graphing the entire line.
To find each intercept, simply substitute and solve within the equation, making the process efficient and straightforward.

x-intercept

The x-intercept is a fundamental aspect of plotting linear equations. By definition, it is the point where the line intersects the x-axis, meaning the value of \(y\) is zero at this point.
Let's find the x-intercept for the equation \(x + y = 6\). Set \(y = 0\) and solve for \(x\):
\[ x + 0 = 6 \] Thus:
\[ x = 6 \]This calculation reveals that the x-intercept occurs at the coordinate point (6, 0). Plot this point on the graph, as it plays a key role in defining the line trajectory.

y-intercept

The y-intercept of a line is equally important as the x-intercept when graphing linear equations. It is the point where the line crosses the y-axis, which means \(x\) must be zero at this location.
For the equation \(x + y = 6\), set \(x = 0\) and solve for \(y\):
\[ 0 + y = 6 \]It leads to:
\[ y = 6 \] Thus, the y-intercept is at (0, 6).
Plot this on the graph, as it's a critical point that, along with the x-intercept, helps define the slope and position of the entire line.

Checkpoint Method

The checkpoint method is an additional step to verify the correctness of your graph. Once you have plotted the intercepts, selecting a third point ensures that the line is accurate. This is done by choosing any point that satisfies the equation and checking if it aligns with the graph.
For the equation \(x + y = 6\), a suitable checkpoint is (3, 3). Substitute these coordinates into the equation:
\[ 3 + 3 = 6 \]Since the equation holds true, (3, 3) is indeed on the line.
Add this point to your graph to form a robust line spanning through all key points, reassuring the accuracy of the drawn line.