Problem 8 - 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. $$4 x-3=2 y$$

Step-by-Step Solution

Answer

The \(x\)-intercept is \(\left(\frac{3}{4}, 0\right)\) and the \(y\)-intercept is \((0, -\frac{3}{2})\).

Step 1: Convert to slope-intercept form

Start by rewriting the equation in the slope-intercept form, which is typically used to find the intercepts easily. Solve for \(y\) to get the equation into the form \(y = mx + b\). Given the equation is \(4x - 3 = 2y\), rearrange to solve for \(y\): \[ 2y = 4x - 3 \] Divide everything by 2: \[ y = 2x - \frac{3}{2} \]

Step 2: Find the y-intercept

The \(y\)-intercept occurs where \(x = 0\). Substitute \(x = 0\) into the derived equation \(y = 2x - \frac{3}{2}\): \[ y = 2(0) - \frac{3}{2} = -\frac{3}{2} \] So, the \(y\)-intercept is at the point \((0, -\frac{3}{2})\).

Step 3: Find the x-intercept

The \(x\)-intercept occurs where \(y = 0\). Substitute \(y = 0\) in the original equation \(4x - 3 = 2y\): \[ 4x - 3 = 0 \] Add 3 on both sides: \[ 4x = 3 \] Divide by 4: \[ x = \frac{3}{4} \] Therefore, the \(x\)-intercept is at the point \((\frac{3}{4}, 0)\).

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is one of the most user-friendly formulas for both plotting and interpreting straight lines. It is expressed as \(y = mx + b\), where:

• \(m\) represents the slope of the line, indicating its steepness or how much \(y\) changes for a unit change in \(x\).
• \(b\) is the y-intercept, defining the point where the line crosses the y-axis.

You can solve for \(y\) in an equation to rearrange it into this form. This setup simplifies the process of finding intercepts and understanding a line's behavior. In this exercise, we started with the equation \(4x - 3 = 2y\). By rearranging, we arrived at \(y = 2x - \frac{3}{2}\). In this new form, we can easily identify both the slope \(2\) and the y-intercept \(-\frac{3}{2}\). Understanding and using the slope-intercept form helps quickly determine key properties of a line.

Finding x-intercept

Finding the x-intercept in a linear equation involves determining the point where the line crosses the x-axis. This happens when \(y\) equals zero because the x-axis is where \(y = 0\). For the equation \(4x - 3 = 2y\), to find the x-intercept, we substitute \(y = 0\) into the equation:

• \(4x - 3 = 0\)
• Solving gives us \(4x = 3\)
• Dividing by 4, we get \(x = \frac{3}{4}\)

The x-coordinate is \(\frac{3}{4}\), and since \(y = 0\), the x-intercept point is \(\left(\frac{3}{4}, 0\right)\). Calculating the x-intercept is useful in graphing because it provides a specific point through which the graph will pass, aiding in drawing an accurate line.

Finding y-intercept

The y-intercept is the point where the line crosses the y-axis, meaning it occurs at \(x = 0\). To determine the y-intercept from the equation \(4x - 3 = 2y\), we first convert the equation into slope-intercept form to ease calculation:

• The equation becomes \(y = 2x - \frac{3}{2}\).
• To find the y-intercept, substitute \(x = 0\):
• \(y = 2(0) - \frac{3}{2} = -\frac{3}{2}\)

Thus, the y-intercept is at the point \(\left(0, -\frac{3}{2}\right)\). Like the x-intercept, the y-intercept is essential for graphing because it provides a way to start drawing the line accurately on a coordinate plane. Knowing the y-intercept quickly allows you to place where the line will vertically intersect the y-axis.

Solving Linear Equations

Solving linear equations involves finding the value of a variable that makes the equation true. Linear equations are expressions of the first degree—essentially a straight line when graphed. This exercise demonstrates finding intercepts by manipulating algebraic equations.When given an equation like \(4x - 3 = 2y\), our first step is to isolate variables to detect when they equal specific values (such as 0 for intercepts). This process might involve:

• Rearranging terms to isolate \(x\) or \(y\).
• Using inverse operations like addition, subtraction, multiplication, or division to simplify the equation.

For instance, to find y when given \(4x - 3 = 0\), if x is known, solve directly. Conversely, when x is input as zero to find the y-intercept, simplifying yields \(y = -\frac{3}{2}\). Solving linear equations methodically breaks down complex problems into small, manageable steps, making intercepts and other properties easy to uncover.