91Ó°ÊÓ

The vertical intercept of a linear equation is the point where the graph of the equation crosses the y-axis. What's special about this point is that the x-coordinate is always zero because it's located directly on the y-axis. To determine the vertical intercept for any equation, simply set \(x = 0\) and solve for \(y\).

For the function given, \(g(x) = 2x + 4\), find the vertical intercept by:

• Substituting \(x = 0\) into the equation: \(g(0) = 2(0) + 4\)
• Solving for \(g(0)\): This simplifies to \(g(0) = 4\)

Thus, the vertical intercept is \((0, 4)\). This means at \(x = 0\), the value of \(g(x)\) or \(y\) is 4. This point, \((0, 4)\), is critical as it can help when quickly graphing linear equations.

The horizontal intercept is where the graph crosses the x-axis. At this point, the y-value (or \(g(x)\)) is zero. This means that the entire function equals zero at the horizontal intercept, influencing the solution of \(x\). To find the horizontal intercept, set \(g(x) = 0\) and solve for \(x\).

For our equation, \(g(x) = 2x + 4\):

• Set \(g(x) = 0\): \(0 = 2x + 4\)
• Subtract 4 from both sides to isolate the term containing \(x\): \(-4 = 2x\)
• Divide both sides by 2: \(x = -2\)

Therefore, the horizontal intercept is \((-2, 0)\), indicating that when the graph crosses the x-axis, the x-coordinate is -2. Recognizing this intercept allows you to quickly sketch where the line intersects the x-axis.

Solving linear equations involves finding values for variables that make the equations true. For linear equations in the form of \(ax + b = 0\), solving them typically includes simple algebraic steps:

1. **Isolate the variable**: Rearrange the equation to get the term with the variable on one side and the other terms on the other side.
2. **Solve for the variable**: Perform arithmetic operations such as addition, subtraction, multiplication, or division to find the value of the variable.

In our example, \(0 = 2x + 4\), these steps are applied as follows:

• Subtract 4 from both sides to isolate the term with \(x\): \(-4 = 2x\).
• Divide each side by 2, simplifying to \(x = -2\).

By methodically handling each step, solving such equations becomes straightforward. With practice, these steps enable you to tackle more complex problems confidently.