Problem 7 - 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. $$ 3 x-2 y=6 $$

Step-by-Step Solution

Answer

x-intercept: (2, 0); y-intercept: (0, -3).

Step 1: Solve for the x-intercept

To find the x-intercept, set \( y = 0 \) and solve for \( x \). Substitute \( y = 0 \) into the equation: \[ 3x - 2(0) = 6 \] This simplifies to: \[ 3x = 6 \] Now, solve for \( x \) by dividing both sides by 3: \[ x = \frac{6}{3} = 2 \] Thus, the x-intercept is at the point (2, 0).

Step 2: Solve for the y-intercept

To find the y-intercept, set \( x = 0 \) and solve for \( y \). Substitute \( x = 0 \) into the equation: \[ 3(0) - 2y = 6 \] This simplifies to: \[ -2y = 6 \] Now, solve for \( y \) by dividing both sides by -2: \[ y = \frac{6}{-2} = -3 \] Thus, the y-intercept is at the point (0, -3).

Key Concepts

x-intercept

The concept of the x-intercept is foundational in understanding how linear equations interact with the x-axis on a graph. The x-intercept is the point where the line crosses the x-axis.
To find the x-intercept, you set the value of y to zero and solve for x. This is because any point on the x-axis has a zero y-value.
For example, given the equation \( 3x - 2y = 6 \), set \( y = 0 \). This converts the equation to \( 3x = 6 \).
Solving for x gives:

• \( x = \frac{6}{3} \)
• \( x = 2 \)

So, the x-intercept is at the coordinate (2, 0). At this point, the line "touches" or "crosses" the x-axis. Understanding how to find the x-intercept provides insights into the behavior of linear equations on a graph.

y-intercept

The y-intercept is where a line crosses the y-axis, and is vital for understanding linear equations' placement on a graph. Any point on the y-axis has an x-value of zero.
To find the y-intercept, we set x to zero in the equation and then solve for y. Using the equation \( 3x - 2y = 6 \), replace x with zero and simplify:

• \( 3(0) - 2y = 6 \)
• This simplifies to \( -2y = 6 \)

By solving for y, we divide both sides by -2:

• \( y = \frac{6}{-2} \)
• \( y = -3 \)

Thus, the y-intercept is at the point (0, -3). Recognizing where a line intercepts the y-axis helps in plotting the line on a Cartesian plane and analyzing its slope and position.

linear equations

Linear equations are equations of degree one, typically in the form \( ax + by = c \). They graph as straight lines with a constant slope.
These equations are the backbone of many mathematical concepts and applications in the real world.
Each linear equation can be characterized by two intercepts:

• The x-intercept
• The y-intercept

The x-intercept occurs when the line crosses the x-axis, while the y-intercept is where the line crosses the y-axis.
By understanding and calculating these intercepts, you can easily formulate and graph the equation.
For the equation \( 3x - 2y = 6 \), identifying these intercepts involves setting either x or y to zero to find the intercepts.
Linear equations play a crucial role in modeling relationships between two variables, allowing us to predict and understand trends.
They are versatile and form the basis for more complex algebraic concepts.