Problem 18 - Algebra 1

Question

Find the \(x\) -intercept of the graph of the equation. $$ x-2 y=6 $$

Step-by-Step Solution

Answer

The x-intercept of the graph of the equation \(x - 2y = 6\) is 6.

Step 1: Set y to 0

The x-intercept is the x-coordinate where the graph of the equation crosses the x-axis, which corresponds to a y-value of 0. The equation provided is \(x - 2y = 6\). To find the x-intercept, set \(y = 0\) into the equation to get \(x - 2*0 = 6\), which simplifies to \(x = 6\).

Step 2: Identify the x-intercept

The value found for x from the simplified equation is the x-intercept. As such, from the equation \(x = 6\), it is evident that the graph of the equation \(x - 2y = 6\) crosses the x-axis at \(x = 6\). This is the x-intercept of the equation.

Key Concepts

Linear Equations

Linear equations are the simplest form of equations that you'll come across in algebra. They can be written in the form of \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants. The beauty of linear equations lies in their straight-line graphs, hence the term 'linear'.

To solve these equations, you usually isolate one variable, generally \(x\) or \(y\), on one side of the equation. For instance, if you have the equation \(x - 2y = 6\), you can solve for \(x\) by adding \(2y\) to both sides if you're interested in the \(x\)-coordinate, or subtract \(x\) and then divide by -2 for the \(y\)-coordinate. When you graph the equation, every point \( (x,y) \) on the line satisfies the original equation, which makes linear equations predictable and straightforward to work with.

Graphing Linear Equations

When graphing linear equations, it's important to find two fundamental points: the \(x\)-intercept and the \(y\)-intercept. The \(x\)-intercept occurs where the line crosses the \(x\)-axis, which means the \(y\)-value is zero at this point. Similarly, the \(y\)-intercept occurs where the line crosses the \(y\)-axis, meaning the \(x\)-value is zero.

Graphing the equation \(x - 2y = 6\) step-by-step, you'd start by finding these intercepts. You've already discovered that the \(x\)-intercept is 6 by setting \(y = 0\) and solving for \(x\). To find the \(y\)-intercept, you'd set \(x = 0\) and solve for \(y\). Once you have these two points, you can draw a straight line through them, which represents all the solutions to the equation. Remember, every point on the line is a solution, showing how X and Y values relate to each other according to the equation.

Solving for X

Solving for \(x\) is a fundamental skill you need to master when working with linear equations. It involves manipulation of the equation to get \(x\) by itself on one side of the equation. Let's use the provided equation \(x - 2y = 6\) as an example.

Since we're finding the \(x\)-intercept, we know that the value of \(y\) will be zero at that point on the graph. So, we replace \(y\) with 0, leading to a simplified equation of \(x = 6\). As you can see, \(x\) is now by itself, and we've solved for \(x\). This value is where the line crosses the \(x\)-axis. This point, called the \(x\)-intercept, is significant because it provides a straightforward method to start graphing a linear equation, or to understand the behavior of the linear relationship represented by the equation.