Problem 24 - Introductory and Intermediate Algebra for College Students 4th

Question

Use intercepts and a checkpoint to graph each equation. $$6 x-2 y=12$$

Step-by-Step Solution

Answer

The x-intercept is 2, the y-intercept is -6, and the checkpoint is (1,-3). These points can be drawn on a graph to represent the equation \( 6x - 2y = 12 \).

Step 1: Calculate the x-intercept

The x-intercept is the point where the line crosses the x-axis. At this point, \( y = 0 \). So, replace \( y \) with 0 in the equation and solve for \( x \). \( 6x - 2(0) = 12 \) simplifies to \( 6x = 12 \) and x therefore equals 2.

Step 2: Calculate the y-intercept

The y-intercept is where the line crosses the y-axis which is when \( x = 0 \). Replace \( x \) with 0 in the equation and solve for \( y \). \( 6(0) - 2y = 12 \) simplifies to \( -2y = 12 \) and y therefore equals -6.

Step 3: Find a checkpoint

Choose a value for \( x \) or \( y \), but not zero because the intercepts have already been calculated. For example, let's choose \( x = 1 \). Then, replace \( x \) with 1 in the equation to solve for \( y \). \( 6(1) - 2y = 12 \) simplifies to \( 6 - 2y = 12 \), then divide by -2 to get \( y = -3 \). So, the point (1, -3) is on the line.

Step 4: Sketch the graph

Draw the x and y-axes, then plot the x-intercept (2, 0), the y-intercept (0, -6) and the checkpoint (1, -3). Draw a straight line through these three points to represent the equation \( 6x - 2y = 12 \)

Key Concepts

Understanding the X-Intercept

When graphing a linear equation, finding the x-intercept is a crucial step. The x-intercept is where the line crosses the x-axis. At this point, the y-coordinate is always zero because the line hasn't moved up or down yet. To find the x-intercept of an equation, set \( y = 0 \) and solve for \( x \). This gives you a point on the line, typically written as \((x, 0)\).

Let's apply this to our example equation, \( 6x - 2y = 12 \). Replace \( y \) with 0:

• \( 6x - 2(0) = 12 \)
• This simplifies to \( 6x = 12 \)
• Dividing both sides by 6 gives \( x = 2 \)

Thus, the x-intercept is the point \((2, 0)\). This tells us that the line will cross the x-axis at \( x = 2 \). When graphing, this point gives us a concrete starting place for our line on the plane.

Exploring the Y-Intercept

The y-intercept is another fundamental element when graphing linear equations. This is the point where the line intersects the y-axis. Here, the x-coordinate is zero because the line hasn't moved left or right yet. To find the y-intercept, set \( x = 0 \) in the equation and solve for \( y \).

Using our equation, \( 6x - 2y = 12 \), we replace \( x \) with 0:

• \( 6(0) - 2y = 12 \)
• This simplifies to \( -2y = 12 \)
• By dividing by -2, we find \( y = -6 \)

Consequently, the y-intercept of the line is \((0, -6)\). When plotting a graph, this point shows where the line touches the y-axis, indicating direction and positioning. The y-intercept gives us another anchor point from which we can sketch the line across the grid.

Using the Checkpoint Method

The checkpoint method is a way to ensure your graph is accurate by using an additional point beyond the intercepts. This involves picking another value for either \( x \) or \( y \), solving for the other variable, and plotting this third point. It acts as a 'check' on the two intercepts, ensuring that they align correctly with the equation.

In our example, with the equation \( 6x - 2y = 12 \), let's choose \( x = 1 \) as a checkpoint:

• Substitute 1 for \( x \): \( 6(1) - 2y = 12 \)
• This simplifies to \( 6 - 2y = 12 \)
• Solving for \( y \) involves rearranging to \( -2y = 6 \)
• Dividing by -2 gives us \( y = -3 \)

So, the checkpoint is \((1, -3)\). Placing this point on the graph and drawing a line through it alongside the intercepts (\((2, 0)\) and \((0, -6)\)) reassures us that our line is correct. The checkpoint method enhances precision in graphing, ensuring checks against mistakes and confirming the line's path.