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The **x-intercept** of a linear equation gives us the point where the line crosses the x-axis on the coordinate plane. To find this point, we set the value of \( y \) to zero in the equation because at the x-intercept point, \( y \) is always zero. For the equation \( 14x - 13y = 1 \), set \( y = 0 \). This simplifies to \( 14x = 1 \). Solving for \( x \), we divide both sides by 14 to get \( x = \frac{1}{14} \). Thus, the x-intercept is at the point \( \left( \frac{1}{14}, 0 \right) \). This tells us that when the line touches the x-axis, it does so at this very specific point.

The **y-intercept** is the point where the line crosses the y-axis on the coordinate plane. Just like the x-intercept, finding the y-intercept involves setting a variable to zero—in this case, the \( x \) value. This is because the y-intercept always has an x-value of zero. For our given equation \( 14x - 13y = 1 \), set \( x = 0 \) to find the y-intercept. The equation simplifies to \(-13y = 1\). Solving for \( y \), we divide both sides by -13, resulting in \( y = -\frac{1}{13} \). Therefore, the y-intercept is \( \left( 0, -\frac{1}{13} \right) \). This point is where the line intersects the y-axis.

The **coordinate plane** is a two-dimensional surface on which we plot points, lines, and curves. It is composed of two perpendicular lines called axes. The horizontal line is the x-axis, and the vertical line is the y-axis. These axes intersect at a point called the origin, which has coordinates \( (0,0) \).

Understanding the coordinate plane is crucial for visualizing linear equations, as each point on this plane has an x-coordinate and a y-coordinate. When dealing with intercepts in linear equations, it’s helpful to plot the points and see how the line interacts with the axes. The coordinate plane allows us to graphically solve equations and observe the relationships between algebraic expressions.

**Solving equations** like our linear equation \( 14x - 13y = 1 \) involves finding the values of the variables that make the equation true. For linear equations, this often includes finding intercepts and graphing the equation on the coordinate plane. Here are a few steps to keep in mind:

• Identify and understand the components of the equation, such as coefficients and variables.
• To find intercepts, substitute appropriate values (usually zero) for one of the variables and solve for the other.
• Simplify the equation where possible to make calculations easier.
• Use algebraic manipulations, such as addition, subtraction, multiplication, or division, to isolate variables and solve for unknowns.

With practice, solving linear equations becomes an intuitive process, helping you to understand how different lines behave on the coordinate plane.