Problem 8 - Prealgebra

Question

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

Step-by-Step Solution

Answer

The ordered pairs are (0,4), (3,0), and (6,-4).

Step 1: Find y when x = 0

To find the value of y when x = 0, substitute x = 0 into the equation. Start with the equation: \[ 4(0) + 3y = 12 \] Simplify to get:\[ 3y = 12 \] Divide by 3:\[ y = 4 \] So, the ordered pair is (0, 4).

Step 2: Find x when y = 0

To find the value of x when y = 0, substitute y = 0 into the equation. Start with the equation: \[ 4x + 3(0) = 12 \] Simplify to get:\[ 4x = 12 \] Divide by 4:\[ x = 3 \] So, the ordered pair is (3, 0).

Step 3: Find x when y = -4

To find the value of x when y = -4, substitute y = -4 into the equation. Start with the equation: \[ 4x + 3(-4) = 12 \] Simplify to get:\[ 4x - 12 = 12 \] Add 12 to both sides:\[ 4x = 24 \] Divide by 4:\[ x = 6 \] So, the ordered pair is (6, -4).

Key Concepts

Understanding Ordered Pairs

An ordered pair is a fundamental concept in graphing linear equations. It consists of two elements representing the coordinates of a point on a plane. Typically, it's written in the format

• \((x, y)\)

where \(x\) is the horizontal coordinate and \(y\) is the vertical coordinate. These pairs provide precise positioning on the grid of a coordinate plane.

For example, if we have the ordered pair \((0, 4)\), it means that the point is located on the y-axis because the x-coordinate is 0. It also tells us that the point is 4 units above the origin. Understanding how to calculate these ordered pairs, especially from an equation, is crucial for accurately plotting points and graphing equations. In the exercise, substitution methods are used to identify missing values and form complete ordered pairs such as \((3, 0)\) and \((6, -4)\). These pairings are essential checkpoints for drawing the accurate graph of the equation.

Exploring X and Y Intercepts

The x and y intercepts are specific points where the graph of an equation crosses the axes of the coordinate plane. These intercepts are crucial in drawing and understanding the linear equation's behavior.

To find the

• y-intercept, set \(x = 0\) and solve for \(y\).

This provides the point where the line crosses the y-axis. The ordered pair formed, such as \((0, 4)\) in our example, is an intercept.

Similarly, to find the

• x-intercept, set \(y = 0\) and solve for \(x\).

This reveals the intersection of the line with the x-axis. The point \((3, 0)\) corresponds to the x-intercept in our scenario. Understanding these intercepts assists in quickly sketching the graph of a linear equation. In addition to the points showing the intercepts, the point \((6, -4)\) is another valid and relevant part of the graph completion.

Navigating the Coordinate Plane

The coordinate plane is a two-dimensional space defined by two perpendicular axes: the x-axis running horizontally and the y-axis running vertically. It is used as a backdrop for plotting points and graphing equations. Each point on this plane is represented by an ordered pair

• \((x, y)\)

The origin, located at \((0, 0)\), is where both axes intersect, serving as the reference point for measuring distance along either axis.

By plotting the ordered pairs obtained from solving an equation, such as \((0, 4)\), \((3, 0)\), and \((6, -4)\), on the coordinate plane, you can draw the line representing the equation \(4x + 3y = 12\). Equip yourself with this knowledge to tackle various exercises, accurately interpret the spatial arrangement of lines, and grasp the interplay between algebra and geometry within this plane.