Problem 37 - Understanding Elementary Algebra with Geometry

Question

In Exercises \(35-46,\) determine which, if any, of the ordered pairs listed satisfy the given equation. $$y=4 x-1 ; \quad(8,7),(7,8),(0,-1)$$

Step-by-Step Solution

Answer

The ordered pair (0, -1) satisfies the equation.

Step 1 - Plug in the first ordered pair (8,7)

To check if the ordered pair (8, 7) satisfies the equation, plug in x=8 and y=7 into the equation y=4x-1. So, substitute these values to get: 7 = 4(8) - 1

Step 2 - Simplify the equation for (8,7)

Simplify the right hand side: 7 = 32 - 1 7 = 31 This is false, so the ordered pair (8, 7) does not satisfy the equation.

Step 3 - Plug in the second ordered pair (7,8)

Next, check the ordered pair (7, 8) by substituting x=7 and y=8 into the equation y=4x-1. Substitute these values to get: 8 = 4(7) - 1

Step 4 - Simplify the equation for (7,8)

Simplify the right hand side: 8 = 28 - 1 8 = 27 This is false, so the ordered pair (7, 8) does not satisfy the equation.

Step 5 - Plug in the third ordered pair (0,-1)

Finally, check the ordered pair (0, -1) by substituting x=0 and y=-1 into the equation y=4x-1. Substitute these values to get: -1 = 4(0) - 1

Step 6 - Simplify the equation for (0,-1)

Simplify the right hand side: -1 = 0 - 1 -1 = -1 This is true, so the ordered pair (0, -1) satisfies the equation.

Key Concepts

ordered pairs

An ordered pair consists of two numbers written in a specific order, usually within parentheses, like (x, y). In these exercises, we need to see if plugging the values of x and y from the ordered pairs into the given equation results in a true statement. For example, given the equation \( y = 4x - 1 \), the ordered pair (8, 7) means x = 8 and y = 7. We substitute these values into the equation and see if both sides of the equation are equal.

substitution method

The substitution method involves replacing the variables in the equation with the values from an ordered pair. This helps us determine if the pair satisfies the equation. Let's go through this step by step:

- Start with the ordered pair (8, 7). Substitute x with 8 and y with 7 into the equation \(y = 4x - 1\). This gives us:
\( 7 = 4(8) - 1 \).

- Simplify the right-hand side of the equation: \( 7 = 32 - 1 \).

- This simplifies to \( 7 = 31 \), which is false. So, (8, 7) does not satisfy the equation.

- Repeat the process for the next ordered pair (7, 8):
\( 8 = 4(7) - 1 \).

- Simplify: \( 8 = 28 - 1 \), which simplifies to \( 8 = 27 \), another false statement.

- Finally, try the ordered pair (0, -1):
\( -1 = 4(0) - 1 \).

- Simplify: \( -1 = 0 - 1 \), which simplifies to \( -1 = -1 \), a true statement, meaning (0, -1) satisfies the equation.

solution verification

To verify the solution, it’s imperative to check both equality and logical consistency of the result. Here’s how you verify step by step:

- Take the original equation \( y = 4x - 1 \) and the previously substituted values.

- For (8, 7): After substitution and simplification, you got \( 7 = 31 \). Since 7 does not equal 31, the ordered pair (8, 7) is incorrect.

- For (7, 8): After substitution and simplification, you got \( 8 = 27 \). Since 8 does not equal 27, the ordered pair (7, 8) is incorrect.

- For (0, -1): After substitution and simplification, you got \( -1 = -1 \). Since both sides are equal, the ordered pair (0, -1) is correct.

This verification process ensures accuracy, confirming that only (0, -1) satisfies the equation \( y = 4x - 1 \). Always double-check your solutions to avoid simple mistakes.