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The x-intercept is a key concept in graphing linear equations. It represents the point where the graph of an equation crosses the x-axis. At this point, the y-coordinate is always zero because the line is touching the x-axis. To find the x-intercept in any linear equation, you simply set the y variable to zero and solve the equation for x.

In the given exercise, the equation provided is \(3x - 4y = 1\). To find the x-intercept, replace the y with zero, making the equation \(3x - 4(0) = 1\), simplifying to \(3x = 1\).

This equation becomes very straightforward to solve: divide both sides by 3. You get \(x = \frac{1}{3}\). This means the x-intercept is represented by the point \(\left( \frac{1}{3}, 0 \right)\). Remember:

• Set \(y = 0\) to find the x-intercept.
• Solve for \(x\).

Knowing how to find the x-intercept helps you interpret and graph linear equations accurately.

The y-intercept is another fundamental aspect when working with linear equations. It is the point where the graph crosses the y-axis. Here, the x-coordinate is zero. The importance of the y-intercept is that it gives one of the easiest points to plot when graphing because one of the values is already known—specifically, \(x = 0\).

In the provided equation \(3x - 4y = 1\), to find the y-intercept, set \(x\) to zero. Replace x in the equation, which gives \(3(0) - 4y = 1\). This simplifies to \(-4y = 1\).

Solve for \(y\) by dividing both sides by -4, resulting in \(y = -\frac{1}{4}\). Thus, the y-intercept of this line is at \(\left(0, -\frac{1}{4}\right)\). Remember:

• Set \(x = 0\) to find the y-intercept.
• Solve for \(y\).

Understanding the y-intercept helps you find another crucial point needed to graph a line on a coordinate plane.

Solving for variables in linear equations is a fundamental skill in algebra. It involves finding the value of the unknown (usually represented as \(x\) or \(y\)) that makes an equation true. The equation typically has variables on one side and constants on the other.

When solving for a specific variable, you want to isolate it on one side of the equation to determine its value. In the case of the exercise with equation \(3x - 4y = 1\), you practiced solving for both \(x\) and \(y\). This involved:

• Setting the opposite variable to zero to find the intercepts.
• Rearranging the equation using basic algebraic operations like addition, subtraction, multiplication, and division.

For example, solving \(3x = 1\) involved dividing both sides by 3 to isolate \(x\). This process is similar for solving \(-4y = 1\), where you divide both sides by -4 to solve for \(y\).

Mastering the technique of solving for variables is essential in algebra, as it provides the basis for finding intercepts, graphing lines, and solving more complex algebraic problems. It allows for translating real-world scenarios into solvable mathematical expressions.