Problem 6 - College Algebra

Question

For each of the following exercises, find the \(x\)-intercept and the \(y\)-intercept without graphing. Write the coordinates of each intercept. $$4 y=2 x-1$$

Step-by-Step Solution

Answer

x-intercept: \(\left(\frac{1}{2}, 0\right)\), y-intercept: \(\left(0, -\frac{1}{4}\right)\).

Step 1: Understanding Intercepts

The task asks for the \(x\)-intercept and the \(y\)-intercept. The \(x\)-intercept of a line is the point where the line crosses the x-axis. This occurs when \(y=0\). The \(y\)-intercept is the point where the line crosses the y-axis, which occurs when \(x=0\). We need to evaluate the given equation \(4y = 2x - 1\) under these conditions.

Step 2: Finding the x-intercept

To find the x-intercept, we set \(y = 0\) in the equation \(4y = 2x - 1\). This gives us:\[4(0) = 2x - 1\]\[0 = 2x - 1\]Adding 1 to both sides, we get \(1 = 2x\). Solving for \(x\), we divide both sides by 2, yielding: \(x = \frac{1}{2}\). So, the x-intercept is \(\left(\frac{1}{2}, 0\right)\).

Step 3: Finding the y-intercept

To find the y-intercept, we set \(x = 0\) in the equation \(4y = 2x - 1\). This gives us:\[4y = 2(0) - 1\]\[4y = -1\]Solving for \(y\), we divide both sides by 4, yielding: \(y = -\frac{1}{4}\). So, the y-intercept is \(\left(0, -\frac{1}{4}\right)\).

Step 4: Verify Results

To verify our results, we substitute the intercept values back into the original equation:For the x-intercept: Substituting \((\frac{1}{2}, 0)\) into \(4y = 2x - 1\) results in \(0 = 2(\frac{1}{2}) - 1\), which simplifies to 0 = 0, confirming correctness.For the y-intercept: Substituting \((0, -\frac{1}{4})\) results in \(4(-\frac{1}{4}) = 2(0) - 1\), which simplifies to -1 = -1, confirming correctness.

Key Concepts

Finding x-intercepts

Finding the x-intercept of a linear equation involves discovering the point at which the line crosses the x-axis. To find this point, you need to set the value of y to zero and solve for x. This is because along the x-axis, the coordinate y is always zero.
In the exercise given, we start with the equation:
\[4y = 2x - 1\]
To find the x-intercept:

• Set \( y = 0 \).
• Substitute it into the equation, resulting in \( 0 = 2x - 1 \).
• Solve for \( x \) by adding 1 to both sides: \( 1 = 2x \).
• Finally, divide by 2 to isolate x: \( x = \frac{1}{2} \).

Hence, the x-intercept is at the point \( \left( \frac{1}{2}, 0 \right) \). It signifies where the line meets the x-axis. Understanding this concept is crucial as x-intercepts often represent the solutions to equations in many real-world scenarios like finding break-even points in business.

Finding y-intercepts

The y-intercept is another critical component of understanding linear equations. It represents the point where a line crosses the y-axis. To determine this intercept, set the value of x to zero and solve the equation for y, since any point on the y-axis has an x-coordinate of zero.
Using the same equation: \[ 4y = 2x - 1 \]
To find the y-intercept:

• Set \( x = 0 \).
• This transforms the equation to \( 4y = -1 \).
• Now, divide each term by 4 to solve for \( y \): \( y = -\frac{1}{4} \).

Thus, the y-intercept is \( (0, -\frac{1}{4}) \). This point is essential in graphing because it gives a starting position from which to draw the line. The y-intercept also provides insight into certain characteristics of the line, like its starting value, when x equals zero.

Linear equations

Linear equations are fundamental in algebra and describe a straight line on a graph. They display relationships between two variables typically denoted as \( x \) and \( y \). A linear equation can be expressed in various forms, one of the most common being the slope-intercept form, \( y = mx + b \). Here, \( m \) is the slope, and \( b \) is the y-intercept.
In our exercise, the equation provided was:\[ 4y = 2x - 1 \]
Although not in the slope-intercept form initially, simple algebraic manipulation can rearrange it into \( y = \frac{1}{2}x - \frac{1}{4} \). From this form, we easily identify:

• The slope is \( \frac{1}{2} \), showing how steep the line is.
• The y-intercept is \( -\frac{1}{4} \), the point where the line crosses the y-axis.

Understanding linear equations is crucial as they form the foundation of more complex algebraic concepts and have widespread applications, such as predicting outcomes and analyzing trends in various fields including science, economics, and engineering.