Problem 11 - Elementary Algebra

Question

Graph the linear equations and inequalities. $$ 2 x=0 $$

Step-by-Step Solution

Answer

Answer: The point on the graph representing the linear equation 2x = 0 is the origin (0,0).

Step 1: Solve for the variable x

To find the value of x that satisfies the given equation, 2x = 0, we need to isolate x. To do this, divide both sides of the equation by 2: $$ x = \frac{0}{2} $$

Step 2: Simplify the equation

Now, we can simplify the equation to find the value of x: $$ x = 0 $$

Step 3: Plot the point on the graph

Since the value of x that satisfies the equation is 0, we can represent this point on the x-axis of a coordinate plane. The point on the graph will be at the origin (0,0) because there is no y-coordinate present.

Key Concepts

Solving Linear Equations

Linear equations are mathematical expressions that form a straight line when graphed on a coordinate plane. The basic idea of solving such equations is to find the value of the variable that satisfies the equation. In our exercise, the linear equation is \(2x = 0\).

• Identify the unknown variable: In this case, it's \(x\).
• Perform a series of operations to isolate the variable: Here, we divided both sides by 2.

The solution to \(2x = 0\) is straightforward. Since any number multiplied by zero results in zero, \(x\) must be zero. Thus, we determine that \(x = 0\). This process of solving linear equations is essential as the foundation for algebra. By mastering these steps, you'll be prepared to tackle more complex equations and inequalities.

Graphing on Coordinate Plane

Graphing is a visual method to represent equations and inequalities on a flat surface called the coordinate plane. A coordinate plane consists of two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical). Each point on this plane can be described with an ordered pair \((x, y)\). When graphing an equation like \(x = 0\), you simply plot the solution on the graph. Since this equation involves only \(x\), it means any value of \(y\) is acceptable.

• The graph will form a vertical line.
• All points on this line will have an x-coordinate of 0.

In our specific exercise, the point (0,0) is located at the origin, where both axes intersect. Understanding graphing on the coordinate plane provides clarity on how solutions operate in a visual manner.

Isolating Variables

Isolating a variable is a key step in solving linear equations. It involves manipulating the equation so that the variable is by itself on one side of the equation. This is crucial to identify the value of the variable explicitly. In the given linear equation \(2x = 0\), the goal is to get \(x\) alone. You achieve this by:

• Removing coefficients from the variable's side.
• Ensuring operations are applied equally to both sides of the equation.

Here, by dividing both sides by 2, we remove the coefficient from \(x\). This simple step makes the equation \(x = 0\) straightforward. Isolating variables is a foundational technique in algebra that helps simplify and solve equations efficiently, paving the way for understanding more advanced mathematical concepts.