Problem 40 - Understanding Elementary Algebra with Geometry

Question

In Exercises \(35-46,\) determine which, if any, of the ordered pairs listed satisfy the given equation. $$-7 x-2 y+20=0 ; \quad(-1,-3),(-3,-1),(2,3)$$

Step-by-Step Solution

Answer

(2, 3) is the only pair that satisfies the equation.

Step 1 - Understand the problem

We need to determine which, if any, of the provided ordered pairs (-1, -3), (-3, -1), and (2, 3) satisfy the given equation: \begin{center}\(-7x - 2y + 20 = 0\)

Step 2 - Substitute the first pair (-1, -3)

Substitute the values of the first ordered pair into the equation: \(-7(-1) - 2(-3) + 20 = 0\) Simplify the equation: \(7 + 6 + 20 = 33 eq 0\) Since the left side does not equal zero, (-1, -3) does not satisfy the equation.

Step 3 - Substitute the second pair (-3, -1)

Substitute the values of the second ordered pair into the equation: \(-7(-3) - 2(-1) + 20 = 0\) Simplify the equation: \(21 + 2 + 20 = 43 eq 0\) Since the left side does not equal zero, (-3, -1) does not satisfy the equation.

Step 4 - Substitute the third pair (2, 3)

Substitute the values of the third ordered pair into the equation: \(-7(2) - 2(3) + 20 = 0\) Simplify the equation: \(-14 - 6 + 20 = 0\) Since the left side equals zero, (2, 3) satisfies the equation.

Key Concepts

ordered pairs

In mathematics, an ordered pair is a pair of elements with a specific order. This means that the position of the elements matters. For example, the ordered pair \((-1, -3)\) is different from \((-3, -1)\). Each element in an ordered pair is taken from a set and represents a specific position.

Ordered pairs are often used to represent coordinates in a two-dimensional space. The first element in the pair corresponds to the x-coordinate, and the second element corresponds to the y-coordinate.
For instance:

• \((-1, -3)\): Here, -1 is the x-coordinate, and -3 is the y-coordinate.
• \((-3, -1)\): Here, -3 is the x-coordinate, and -1 is the y-coordinate.
• \( (2, 3) \): Here, 2 is the x-coordinate, and 3 is the y-coordinate.

Ordered pairs are essential for solving linear equations, as they provide specific values to substitute into the equation.

This helps to determine if the ordered pair satisfies the equation.

substitution method

The substitution method is a technique used to solve equations by substituting the values of variables into the equation.
This method helps to find whether specific ordered pairs are solutions to an equation.

Here's how you can use the substitution method step-by-step:

• Identify the given equation and the ordered pairs.
• Substitute the x-coordinate and y-coordinate from the ordered pair into the equation.
• Simplify the equation to check if the left-hand side equals the right-hand side.

Let's illustrate with examples:

For the ordered pair \((-1, -3)\) and the equation \( -7x - 2y + 20 = 0 \):
\(-7(-1) - 2(-3) + 20 = 0 \)
Simplify: \( 7 + 6 + 20 = 33 \) — Since 33 is not equal to 0, \((-1, -3)\) is not a solution.

For the ordered pair \((-3, -1)\):
\(-7(-3) - 2(-1) + 20 = 0 \)
Simplify: \(21 + 2 + 20 = 43 \) — Since 43 is not equal to 0, \((-3, -1)\) is not a solution.

Finally, for the ordered pair \((2, 3)\):
\(-7(2) - 2(3) + 20 = 0 \)
Simplify: \(-14 - 6 + 20 = 0 \) — Since the left-hand side equals 0, \((2, 3)\) is a solution.

By substituting values and simplifying, you can determine which ordered pairs satisfy the equation.

simplifying algebraic expressions

Simplifying algebraic expressions involves combining like terms to make the expression easier to work with.
It is an essential skill in solving linear equations, as it helps to reveal the solution more clearly.

Let's break down the process of simplifying:

• Combine like terms: Terms that have the same variable raised to the same power.
• Perform arithmetic operations: Add, subtract, multiply, or divide the coefficients of the terms.

For example, to simplify the expression \( -7(-1) - 2(-3) + 20 \):

Start with the given expression:
\( -7(-1) - 2(-3) + 20 \)

Apply arithmetic operations:
\(-7(-1) = 7 \)
\(-2(-3) = 6 \)
7 + 6 + 20 = 33

By performing these operations, you simplify the expression to get the final result.

Simplifying equations helps clarify whether an equation is true or false for specific ordered pairs.
It reduces complex expressions to simpler forms, making the solving process more straightforward.
Remember to always combine like terms and perform arithmetic operations carefully to avoid mistakes.