Problem 53 - Understanding Elementary Algebra with Geometry

Question

Given the equation \(3 x+2 y=12,\) complete the given ordered pairs: $$(2, \quad) (0, \quad) \quad(\quad,-3) \quad(\quad, 0)$$

Step-by-Step Solution

Answer

The ordered pairs are (2, 3), (0, 6), (6, -3), and (4, 0).

Step 1 - Solve for y when x = 2

Substitute x = 2 into the equation: 3(2) + 2y = 12 6 + 2y = 12 Subtract 6 from both sides: 2y = 6 Divide by 2: y = 3 The ordered pair is (2, 3).

Step 2 - Solve for y when x = 0

Substitute x = 0 into the equation: 3(0) + 2y = 12 0 + 2y = 12 Divide by 2: y = 6 The ordered pair is (0, 6).

Step 3 - Solve for x when y = -3

Substitute y = -3 into the equation: 3x + 2(-3) = 12 3x - 6 = 12 Add 6 to both sides: 3x = 18 Divide by 3: x = 6 The ordered pair is (6, -3).

Step 4 - Solve for x when y = 0

Substitute y = 0 into the equation: 3x + 2(0) = 12 3x + 0 = 12 Divide by 3: x = 4 The ordered pair is (4, 0).

Key Concepts

Ordered Pairs

Ordered pairs are a fundamental concept in coordinate geometry. They consist of two values arranged in a specific order within parentheses, like \( (x, y) \). The first value represents the x-coordinate, and the second value represents the y-coordinate.

To complete an ordered pair, we need to find the missing value using the given linear equation, like in our example: \(3x + 2y = 12\). By substituting one known value, we can solve for the other. For instance, if we are given the pair \( (2, \_ \) and need to find y, we substitute \( x = 2 \) into the equation to solve for y.

Substitution Method

The substitution method is a technique to solve systems of equations or to find missing variables within an equation by substituting known values into the equation.

Here is a simple step-by-step approach to using substitution in our equation \(3x + 2y = 12\):

• Identify the known value in the ordered pair.
• Substitute this value into the equation.
• Solve for the remaining variable.

For example, when given \( (x , -3) \), we substitute y = -3 to find x: \[ 3x + 2(-3) = 12 \]
Simplify and solve for x.

Algebraic Manipulation

Algebraic manipulation involves using arithmetic operations to solve equations. This includes adding, subtracting, multiplying, or dividing both sides of an equation to isolate a variable.

In our example, algebraic manipulation helps to solve for x or y:

• For \( 3x + 2(0) = 12 \), we simplified to \( 3x = 12 \) using subtraction and division.
• Similarly, \(3(0) + 2y = 12 \) becomes \( 2y = 12 \).

Manipulating equations step by step ensures accuracy and clarity in finding the solution.

Linear Equations

A linear equation is an equation between two variables that produces a straight line when plotted on a graph. It typically takes the form \(Ax + By = C\).

The general process for solving linear equations includes:

• Identifying the given values or ordered pairs.
• Substituting these values into the equation.
• Using algebraic manipulation to find the missing variable.

In our example, \(3x + 2y = 12 \) is a linear equation. By following these steps, we can complete the ordered pairs such as \( (2, 3) \) and \( (6, -3) \).