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The x-intercept of a graph is a point where the line crosses the x-axis. Understanding how to find the x-intercept is crucial for graphing linear equations.

To find the x-intercept, you set the value of y equal to zero in the given equation. This is because the x-intercept occurs where the line meets the x-axis, and at every point on the x-axis, the y-coordinate is zero. Let's look at the equation from our problem: \(6x - 2y - 12 = 0\).

When you plug in \(y = 0\), the equation simplifies to \(6x - 12 = 0\). By solving for x, we get:

• Add 12 to both sides: \(6x = 12\)
• Divide both sides by 6: \(x = 2\)

Thus, the x-intercept is the point \((2, 0)\). This means our line crosses the x-axis at the coordinate \((2, 0)\). Always remember that the x-intercept will be expressed as \((a, 0)\), with 'a' being the calculated x-value.

The y-intercept is the point where the line crosses the y-axis. Finding the y-intercept involves setting the x-coordinate to zero because, on the y-axis, the x-coordinate is always zero. Let's find the y-intercept for our equation \(6x - 2y - 12 = 0\).

Substitute \(x = 0\) into the equation. This results in \(-2y - 12 = 0\). Solving for y is straightforward:

• Add 12 to both sides: \(-2y = 12\)
• Divide both sides by -2: \(y = -6\)

Therefore, the y-intercept is \((0, -6)\). The line will cross the y-axis at this point \((0, -6)\). It's typically written in the form of \((0, b)\) with 'b' being the calculated y-value.

These intercepts, both x and y, provide crucial points that make graphing linear equations straightforward.

Equation solving is a fundamental skill in algebra that involves finding values for variables that make an equation true. Let's explore this concept using our example equation, \(6x - 2y - 12 = 0\), focusing on determining intercepts which are key for graphing.

Solving for intercepts involves:

• Identifying which variable to set as zero (x for y-intercept and y for x-intercept).
• Substituting this value back into the equation simplifies it to find the other variable.
• Carrying out basic arithmetic operations to isolate the variable you're solving for.

In our exercise:For the x-intercept, setting \(y = 0\) allowed us to solve \(6x - 12 = 0\), isolating x as shown above. Similarly, setting \(x = 0\) to solve for the y-intercept gave us \(-2y - 12 = 0\). This approach simplifies equations, yielding easy solutions for intercepts, which is vital not only for solving equations but also for tasks such as graphing.

Remember, breaking down problems into small steps can reduce errors and build confidence in solving more complex equations.