Problem 74 - Precalculus Essentials

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The ordered pair (2,5) satisfies \(3 y-2 x=-4\)

Step-by-Step Solution

Answer

The statement is false. The correct equation that the ordered pair (2,5) does satisfy should be \(3y - 2x = 11\).

Step 1: Identify the ordered pair

The given ordered pair is (2,5). This means \(x=2\) and \(y=5\).

Step 2: Substitute the ordered pair into the equation

Substitute \(x=2\) and \(y=5\) into the equation \(3y-2x = -4\), so it becomes \(3(5) - 2(2)\).

Step 3: Solve the equation

The equation becomes \(15 - 4 = 11\).

Step 4: Determine whether the statement is true or false

The equation \(3y - 2x = -4\) with the ordered pair (2,5) yields 11, not -4. Therefore, the statement that the ordered pair \((2,5)\) satisfies the equation is false.

Step 5: Adjust the equation to make the statement true

The original equation \(3y - 2x = -4\) should be adjusted to \(3y - 2x = 11\) to make the statement true with the ordered pair (2,5).

Key Concepts

Ordered Pairs

When we talk about ordered pairs in algebra, we are referring to combinations of two values that usually represent coordinates on a plane.
The first value is the x-coordinate, and the second value is the y-coordinate. This pair is often written as \(x, y\).
In the context of solving linear equations, ordered pairs play a crucial role as they help us determine if specific values satisfy a given equation. For example, if we have an ordered pair (2,5), this tells us that \(x = 2\) and \(y = 5\). To determine if this pair satisfies an equation like \(3y - 2x = -4\), we substitute these values into the equation.

• First coordinate (x) aligns with the horizontal position or input variable.
• Second coordinate (y) aligns with the vertical position or output variable.
• Substitution of coordinates into equations helps verify whether the equation holds true.

Understanding how to interpret and use ordered pairs is fundamental in solving equations and analyzing mathematical relationships.

Substitution Method

The substitution method is a basic technique used to solve equations in math.
It's especially useful when working with ordered pairs.
The process involves replacing variables in an equation with specific values to determine if those values satisfy the equation. Let's break down how this works:1. **Identify the Variables**: Start by identifying which values correspond to \(x\) and \(y\) in your ordered pair. For instance, in (2,5), \(x = 2\) and \(y = 5\).2. **Substitute the Values**: Plug these values into the equation. If our equation is \(3y - 2x = -4\), replace \(y\) with 5 and \(x\) with 2, transforming the equation to \(3(5) - 2(2)\).3. **Calculate**: Perform the arithmetic to either confirm or refute if the equation holds true. Here, we'd calculate \(3 \times 5 = 15\) and \(2 \times 2 = 4\), making it \(15 - 4\). This results in 11, not -4.Using substitution not only helps verify potential solutions but also sharpens understanding of equation dynamics.

• Substitution converts abstract symbols into actionable values.
• Ensures clarity on how variables interact within equations.
• Facilitates quick checks of ordered pair validity against given equations.

True or False Statements

Mathematics often requires determining the truthfulness of statements, particularly whether a given ordered pair satisfies an equation.
We call this analysis a process of evaluating true or false statements.
In our exercise, the statement is that the ordered pair (2,5) satisfies the equation \(3y - 2x = -4\).To ascertain the statement's truth, we compare the results of substitution (using the method outlined earlier) against the original equation outcome. If the values match the equation outcome, the statement is true.

• Conduct a calculation post substitution to see if the result equals the equation’s right side.
• Match and verify the resulting value to assess the truth of the statement.

Here, our proposed pair results in 11 rather than -4, confirming the statement is false. In the context of solving equations, this process helps us make necessary adjustments or corrections to align the statement with mathematical reality.