Problem 9 - College Algebra with Corequisite Support

Question

For each of the following exercises, find the \(x\) -intercept and the \(y\) -intercept without graphing. Write the coordinates of each intercept. $$ 3 x+8 y=9 $$

Step-by-Step Solution

Answer

x-intercept: (3, 0), y-intercept: (0, 9/8)

Step 1: Understand Intercepts

The x-intercept is where the graph of the equation crosses the x-axis. To find it, we set \( y = 0 \) and solve for \( x \). The y-intercept is where the graph crosses the y-axis. To find it, we set \( x = 0 \) and solve for \( y \).

Step 2: Find the x-intercept

Set \( y = 0 \) in the equation \( 3x + 8y = 9 \). This gives us: \( 3x + 8(0) = 9 \), which simplifies to \( 3x = 9 \). Solving for \( x \), we get \( x = \frac{9}{3} = 3 \). Thus, the x-intercept is at the point \((3, 0)\).

Step 3: Find the y-intercept

Set \( x = 0 \) in the equation \( 3x + 8y = 9 \). This gives us: \( 3(0) + 8y = 9 \), which simplifies to \( 8y = 9 \). Solving for \( y \), we get \( y = \frac{9}{8} \). Thus, the y-intercept is at the point \((0, \frac{9}{8})\).

Step 4: Compile Results

The x-intercept is \((3, 0)\) and the y-intercept is \((0, \frac{9}{8})\). These are the points where the line crosses the x-axis and y-axis, respectively.

Key Concepts

Understanding the X-Intercept

The x-intercept is a crucial point on the graph of a linear equation. It is the place where the graph crosses the x-axis. To find the x-intercept, we make the equation simpler by setting the y-value to zero.

Why is this so? Because at any point on the x-axis, the y-value is zero. This makes calculations straightforward. For instance, with the equation \( 3x + 8y = 9 \), substituting \( y = 0 \) transforms the equation to \( 3x = 9 \).

• Simplify to find: \( 3x = 9 \)
• Divide both sides by 3 to solve for x
• Thus, \( x = \frac{9}{3} = 3 \)

Consequently, the x-intercept for our equation sits at the coordinate \((3, 0)\). Grasping this concept helps in understanding how equations relate to their graphical representations.

Discovering the Y-Intercept

The y-intercept, similar to the x-intercept, marks where the line cuts through the y-axis on a graph. Here, we aim to find the corresponding y-value by setting x to zero.

Again, the logic follows just like with the x-axis: any point found directly on the y-axis has an x-value of zero. So, in the equation \( 3x + 8y = 9 \), we substitute \( x = 0 \).

• The equation becomes: \( 8y = 9 \)
• Divide each side by 8
• This gives \( y = \frac{9}{8} \)

Thus, the coordinate for the y-intercept is \((0, \frac{9}{8})\). Knowing the y-intercept is beneficial when plotting linear equations, ensuring a reliable starting point to draw the line accurately.

The Art of Solving Linear Equations

Solving linear equations is essential for algebra. It involves finding the value of unknown variables that make an equation true. Linear equations follow the standard form \( ax + by = c \), where a, b, and c are constants.

To tackle such equations step-by-step:

• Identify the type of equation you are working with
• Use known values or change the equation's form to isolate unknowns
• Perform algebraic operations, like addition, subtraction, multiplication, or division

For understanding intercepts, we adjust our equation slightly by setting one of the variables to zero. This action isolates the other variable, making the path to the solution clearer.

Mastering solving linear equations equips students with tools to approach more complex mathematical problems efficiently.