Problem 6 - 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. $$ 4 y=2 x-1 $$

Step-by-Step Solution

Answer

The x-intercept is \((\frac{1}{2}, 0)\) and the y-intercept is \((0, \frac{-1}{4})\).

Step 1: Identifying the Intercepts

To find the intercepts of a linear equation, we find where the graph of the equation crosses the axes. The x-intercept occurs where the line crosses the x-axis (y = 0), and the y-intercept occurs where it crosses the y-axis (x = 0).

Step 2: Finding the y-intercept

To find the y-intercept, set the value of x to 0 in the equation. The y-intercept is the point where the line crosses the y-axis (x=0).Start with the equation:\[ 4y = 2(0) - 1 \]Simplify it:\[ 4y = -1 \]Solve for y by dividing both sides by 4:\[ y = \frac{-1}{4} \]The y-intercept is \((0, \frac{-1}{4})\).

Step 3: Finding the x-intercept

To find the x-intercept, set the value of y to 0 in the equation. The x-intercept is the point where the line crosses the x-axis (y=0).Start with the equation:\[ 4(0) = 2x - 1 \]Simplify it:\[ 0 = 2x - 1 \]Add 1 to both sides to isolate the term with x:\[ 1 = 2x \]Divide both sides by 2 to solve for x:\[ x = \frac{1}{2} \]The x-intercept is \((\frac{1}{2}, 0)\).

Key Concepts

Intercepts

Intercepts in a linear equation help us understand where the graph of that equation meets the axes on a coordinate plane. These points are crucial since they provide a clear picture of the line's behavior.

An intercept is a point where the line intersects either the x-axis (the horizontal line) or the y-axis (the vertical line).

• The x-intercept is the point where the line crosses the x-axis.
• The y-intercept is the point where the line crosses the y-axis.

By understanding these points, even without drawing a graph, you can gain a lot of insights into the characteristics of a linear equation.

For instance, in the equation, you simply set one variable to zero and solve for the other to find the intercepts.

X-Intercept

The x-intercept of a linear equation is the point where the graph crosses the x-axis. At this point, the y-coordinate is always zero.

To find the x-intercept, follow these simple steps:

• Set the y variable to zero in the equation of the line.
• Solve the resulting equation for x.

Given the equation: 4y = 2x - 1,

Setting y = 0, we get: 4(0) = 2x - 1.
This simplifies to 0 = 2x - 1.
Solving for x, we add 1 to both sides, resulting in 1 = 2x,
and then divide both sides by 2 to find: x = \( \frac{1}{2} \).

Thus, the x-intercept is \( (\frac{1}{2}, 0) \), which means the line meets the x-axis at the point \( (\frac{1}{2}, 0) \).

Y-Intercept

The y-intercept of a linear equation is the point where the graph crosses the y-axis. Here, the x-coordinate is zero.

Finding the y-intercept involves:

• Setting the x variable to zero in the equation.
• Solving for y to find the intersection point.

Consider the same equation: 4y = 2x - 1.

Setting x = 0, the equation becomes: 4y = 2(0) - 1.
Simplifying it gives: 4y = -1.
Divide both sides by 4 to isolate y: y = \( \frac{-1}{4} \).

So, the y-intercept is \( (0, \frac{-1}{4}) \). This indicates the line passes through the point \( (0, \frac{-1}{4}) \) on the y-axis.