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

Step-by-Step Solution

Answer

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

Step 1: Understand the Equation

The given equation is \(4x - 3 = 2y\). Our goal is to find the \(x\)-intercept and \(y\)-intercept from this equation.

Step 2: Substitute to Find the x-Intercept

To find the \(x\)-intercept, substitute \(y = 0\) into the equation:\[ 4x - 3 = 2(0) \]This simplifies to:\[ 4x - 3 = 0 \]

Step 3: Solve for x to Get the x-Intercept

Solve the equation \(4x - 3 = 0\) to find \(x\):\[ 4x = 3 \]Divide by 4:\[ x = \frac{3}{4} \]Thus, the \(x\)-intercept is \(\left( \frac{3}{4}, 0 \right)\).

Step 4: Substitute to Find the y-Intercept

To find the \(y\)-intercept, substitute \(x = 0\) into the original equation:\[ 4(0) - 3 = 2y \]This simplifies to:\[ -3 = 2y \]

Step 5: Solve for y to Get the y-Intercept

Solve the equation \(-3 = 2y\) to find \(y\):\[ y = -\frac{3}{2} \]Thus, the \(y\)-intercept is \((0, -\frac{3}{2})\).

Key Concepts

Understanding the x-Intercept

The x-intercept of a line is the point where the line crosses the x-axis. At this point, the value of the y-coordinate is always zero. To find the x-intercept, we set y = 0 in the equation and solve for x. This will give us the coordinate of the x-intercept in the form \((x, 0)\).

Let's consider the equation from our problem: \(4x - 3 = 2y\). To find the x-intercept, replace y with 0 and solve the resulting equation, \(4x - 3 = 0\).

• Solve for x: add 3 to both sides to get \(4x = 3\).
• Next, divide both sides by 4 to isolate x, resulting in \(x = \frac{3}{4}\).

Thus, the x-intercept is \(\left( \frac{3}{4}, 0 \right)\), which means the line crosses the x-axis at \(x = \frac{3}{4}\).

Finding the x-intercept is useful because it tells us how the line behaves as it approaches the x-axis.

Understanding the y-Intercept

The y-intercept of a line is the point where the line crosses the y-axis. Here, the x-coordinate is equal to zero, since at the intersection with the y-axis, the x-value is always zero. To find the y-intercept, we substitute \(x = 0\) into the equation and solve for y to get the coordinate \((0, y)\).

For our equation, \(4x - 3 = 2y\), we substitute 0 for x and simplify the equation: \(-3 = 2y\).

• Solve for y: divide both sides by 2, resulting in \(y = -\frac{3}{2}\).

Therefore, the y-intercept is \((0, -\frac{3}{2})\). This tells us that the line crosses the y-axis at \(-\frac{3}{2}\).

Knowing the y-intercept helps to understand how the line interacts with the vertical axis and is crucial in graphing the line accurately.

Exploring Linear Equations

Linear equations represent straight lines when graphed on the coordinate plane. They often appear in the form \(ax + by = c\), where a, b, and c are constants, and x and y are variables. Understanding linear equations allows us to find intercepts and better grasp the line's behavior.

The given equation \(4x - 3 = 2y\) is a linear equation. It can be rearranged to the standard form of a linear equation by simply moving terms around, or it can also be manipulated into the slope-intercept form \(y = mx + b\), where m is the slope, and b is the y-intercept.

• To convert: first isolate y by moving terms: \(2y = 4x - 3\).
• Divide every term by 2 to solve for y: \(y = 2x - \frac{3}{2}\).

This reveals the slope \(m = 2\) and confirms the y-intercept \(b = -\frac{3}{2}\). The slope tells us how steep the line is, while intercepts provide specific points where the line crosses the axes.

Understanding these components in linear equations makes it easier to sketch graphs and predict the line's behavior in various situations.