Problem 30 - Elementary Algebra

Question

For the following problems, graph the equations. $$ -3 x=-9 $$

Step-by-Step Solution

Answer

Answer: The graph of the equation -3x = -9 is a vertical line passing through the point (3,0) on the coordinate plane.

Step 1: Solve for x

To graph the equation \(-3x = -9\), we first need to isolate the variable x. Divide both sides of the equation by \(-3\) to get the value of x: $$ \frac{-3x}{-3} = \frac{-9}{-3} \\ x = 3 $$

Step 2: Plot the value of x on the graph

Now that we have the value of x, which is \(3\), we can make a plot on the graph. Since the equation only involves the x variable, the line will be a vertical line on the coordinate plane. The line will pass through the point \((3,0)\), which is the x-intercept, meaning all points on the line will have an x-coordinate of \(3\).

Step 3: Draw the vertical line

Lastly, draw a vertical line passing through the point \((3,0)\), extending both up and down from the x-axis. This line represents the graph of the equation \(-3x = -9\).

Key Concepts

Solving Linear Equations

Solving linear equations is like unraveling a puzzle where the goal is to find the value of the unknown variable. In the example provided, the linear equation is \(-3x = -9\). This is a simple one-variable equation. To solve it, you need to get "\(x\)" by itself on one side of the equation.

Here's how to approach it step-by-step:

• Identify the equation you need to solve. In this case, \(-3x = -9\).
• To isolate \(x\), divide both sides of the equation by the coefficient of \(x\), which is \(-3\).
• Once you do the division, you get \(x = 3\).

After solving for \(x\), you've found the value of the variable. Solving linear equations is a fundamental skill in algebra and provides the basis for understanding more complex equations later.

Coordinate Plane

The coordinate plane is a two-dimensional surface where points can be expressed in terms of coordinates. Each point on this plane is represented by an ordered pair \((x, y)\).

Here's how the coordinate plane works:

• The horizontal axis is called the x-axis.
• The vertical axis is called the y-axis.
• The point where the axes intersect is called the origin, represented by \((0, 0)\).

Any point on the plane can be located by moving along the x-axis to the correct x-coordinate, and then moving up or down to the correct y-coordinate. For our example, after finding that \(x = 3\), the point on the coordinate plane relating to it would be \((3, 0)\), because there's no y-component in the equation.

Vertical Line

A vertical line on a coordinate plane means that all points on the line have the same x-coordinate. In the case of the equation \(-3x = -9\), once solved, it gives us \(x = 3\). This tells us that no matter the value of \(y\), the x-coordinate will always be 3.

To draw a vertical line:

• Mark the point \((3, 0)\) on the x-axis; this is your x-intercept.
• Extend a line straight up and down through \((3, 0)\). This line is parallel to the y-axis.

Vertical lines are unique because they have an undefined slope, as they literally go straight up and down. Remember, the equation \(x = a\) produces a vertical line that crosses the x-axis at the point \(a\). This is a crucial concept in graphing linear equations and understanding the nature of different line equations.