Problem 10 - Prealgebra

Question

Complete the given ordered pairs, and use the results to graph the equation. (GRAPH CANT COPY) $$4 x-3 y=12 \quad(0, \quad),(, 0),(, 4)$$

Step-by-Step Solution

Answer

The complete ordered pairs are (0, -4), (3, 0), and (4, \frac{4}{3}).

Step 1: Solve for y when x = 0

Substitute \( x = 0 \) into the equation \( 4x - 3y = 12 \). This gives \( 4(0) - 3y = 12 \), which simplifies to \( -3y = 12 \). Divide both sides by -3 to get \( y = -4 \). Therefore, the ordered pair is \( (0, -4) \).

Step 2: Solve for x when y = 0

Substitute \( y = 0 \) into the equation \( 4x - 3y = 12 \). This gives \( 4x - 3(0) = 12 \), which simplifies to \( 4x = 12 \). Divide both sides by 4 to get \( x = 3 \). Therefore, the ordered pair is \( (3, 0) \).

Step 3: Solve for y when x = 4

Substitute \( x = 4 \) into the equation \( 4x - 3y = 12 \). This gives \( 4(4) - 3y = 12 \), which simplifies to \( 16 - 3y = 12 \). Subtract 16 from both sides to get \( -3y = -4 \). Divide both sides by -3 to get \( y = \frac{4}{3} \). Therefore, the ordered pair is \( (4, \frac{4}{3}) \).

Step 4: Graph the Equation Using Ordered Pairs

Now, plot the three ordered pairs \((0, -4)\), \((3, 0)\), and \((4, \frac{4}{3})\) on the coordinate plane. Draw a straight line through these points to graph the equation \(4x - 3y = 12\). This line represents all solutions to the equation.

Key Concepts

Ordered Pairs

Ordered pairs are a fundamental concept in mathematics, especially when dealing with graphing linear equations. An ordered pair consists of two numbers: the first represents the x-coordinate, and the second represents the y-coordinate. This is usually written in the form \(x, y\). The order is crucial because it determines the exact point's position on the coordinate plane.

For example, in the equation \(4x - 3y = 12\), solving for ordered pairs involves finding values for x and y that hold true to the equation. If we substitute specific values for x or y, we can solve for the other variable. This allows us to accurately identify points on a graph. By substituting x = 0, y = 0, and x = 4 into the equation, we found the ordered pairs \(0, -4\), \(3, 0\), and \(4, \frac{4}{3}\) respectively. Each represents a specific location on the graph that satisfies the equation.

Coordinate Plane

The coordinate plane is a two-dimensional surface on which we can plot points, lines, and curves. It has two axes: the horizontal axis (x-axis) and the vertical axis (y-axis). These axes intersect at a point called the origin, denoted as (0,0).

To understand how the coordinate plane works:

• The x-coordinate indicates a point's horizontal position relative to the origin.
• The y-coordinate indicates the vertical position relative to the origin.

Using the coordinate plane, you can visually represent the solutions to algebraic equations. In our problem, once we find ordered pairs like \(0, -4\), \(3, 0\), and \(4, \frac{4}{3}\), we plot them on this plane to form the line represented by the equation \(4x - 3y = 12\). This helps us see all possible solutions that form a straight line.

Solving for Variables

Solving for variables is the process of isolating a variable in an equation to find its value. This is a core algebraic skill essential for graphing equations. The equation given is \(4x - 3y = 12\). To find ordered pairs, we substitute known values for x or y and solve for the other variable.

Here's how each variable is tackled:

• Substitute x = 0: Solving \(-3y = 12\) results in \(y = -4\).
• Substitute y = 0: Solving \(4x = 12\) results in \(x = 3\).
• Substitute x = 4: Solving \(-3y = -4\) results in \(y = \frac{4}{3}\).

This approach helps us systematically find points that lie on the line, making it easier to graph these mathematical relationships.

Plotting Points

Plotting points is the art of placing points on the coordinate plane, based on their ordered pair values. It requires a clear understanding of both the x and y coordinates to accurately represent their position.

Here's a simple way to plot points:

• Locate the x-coordinate on the horizontal line (x-axis).
• From this position, go along the vertical path until you meet the y-coordinate.
• Mark this intersection as a point on the plane.

For the equation \(4x - 3y = 12\), when you've determined the ordered pairs, such as \(0, -4\), \(3, 0\), and \(4, \frac{4}{3}\), you can efficiently plot them. Once plotted, draw a straight line through these points. Each point represents a solution to the equation, together forming the graph of the line.