Problem 41 - Understanding Elementary Algebra with Geometry

Question

In Exercises \(35-46,\) determine which, if any, of the ordered pairs listed satisfy the given equation. $$\frac{2}{3} x-\frac{1}{2} y=1 ; \quad(6,2),(-6,-10),(3,-2)$$

Step-by-Step Solution

Answer

(6,2) and (-6,-10) satisfy the equation.

Step 1 - Substitute the first pair \(6,2\)

To check if the ordered pair \(6,2\) satisfies the equation, substitute \(x = 6\) and \(y = 2\) into the equation \frac{2}{3}x - \frac{1}{2}y = 1\: \[ \frac{2}{3}(6) - \frac{1}{2}(2) = 1 \] Simplify the left side: \[ 4 - 1 = 1 \] Which is true, so \(6,2\) satisfies the equation.

Step 2 - Substitute the second pair \((-6,-10)\)

Next, substitute \(x = -6\) and \(y = -10\) into the equation: \[ \frac{2}{3}(-6) - \frac{1}{2}(-10) = 1 \] Simplify the left side: \[ -4 + 5 = 1 \] Which is true, so \((-6,-10)\) satisfies the equation.

Step 3 - Substitute the third pair \(3,-2\)

Finally, substitute \(x = 3\) and \(y = -2\) into the equation: \[ \frac{2}{3}(3) - \frac{1}{2}(-2) = 1 \] Simplify the left side: \[ 2 + 1 = 3 \] Which is false, so \(3,-2\) does not satisfy the equation.

Key Concepts

substitution method

The substitution method is a technique used to determine if an ordered pair is a solution of a given equation. Here, we substitute the values of the ordered pair into the equation and simplify.

This involves several steps:

• Identify the ordered pair to be tested. For example, let's use the pair (6, 2).
• Replace the variables in the equation with the values from the ordered pair. So, if our equation is \(\frac{2}{3}x - \frac{1}{2}y = 1\), we substitute \(x = 6\) and \(y = 2\)
.

This process helps to determine if the resulting statement is true or false.
If true, the pair is a solution; if false, it is not.

ordered pairs

An ordered pair consists of two elements, usually represented as (x, y). These pairs can represent coordinates on a graph, or solutions to equations.

For example, (6, 2) means \(x = 6\) and \(y = 2\).

Using ordered pairs, we can test multiple pairs on a single equation to determine which pairs satisfy the equation.

• The first number is the x-coordinate, and the second is the y-coordinate.
• Each ordered pair is unique for each equation.
• Ordered pairs are essential in systems of equations as they help in finding common solutions.

Understanding how to use and test ordered pairs is a fundamental skill in algebra.

simplifying equations

Simplifying equations is a crucial step in solving any mathematical problem. It involves reducing the equation to its simplest form to easily interpret and solve.

For instance, if we start with \( \frac{2}{3}(6) - \frac{1}{2}(2) = 1 \), we first perform the multiplications:

• Calculate \( \frac{2}{3} \times 6 = 4 \)
• Calculate \( \frac{1}{2} \times 2 = 1 \)
• So, the equation simplifies to \( 4 - 1 = 1 \), which is true.

By following these steps, you confirm accuracy and correctness.
Each simplification should bring you closer to understanding if the ordered pair satisfied the equation.
Always remember to perform each arithmetic operation carefully!