Problem 74 - College Algebra Essentials

Question

Determine whether each statement is true or Jalse. 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 ordered pair (2,5) does not satisfy the equation \(3y-2x=-4\)

Step 1: Identify the 'x' and 'y' values from the provided ordered pair

From the provided pair \((2,5)\) 'x' stands for the first value of the pair which is 2 and 'y' stands for the second value of the pair which is 5.

Step 2: Substitute x and y into the given equation

Substitute \(x = 2\) and \(y = 5\) into the equation \(3y - 2x = -4\). It becomes: \(3 * 5 - 2 * 2 = -4\).

Step 3: Solve the equation

After combining similar terms, the left-hand side \(3*5 - 2*2\) simplifies to 15 - 4 = 11.

Step 4: Check whether the sides of the equation are equal

After solving both sides, we get: left-hand side = 11, right-hand side = -4. Since the two sides are not equal, the provided statement is false.

Key Concepts

Ordered Pairs

An ordered pair is a fundamental concept in mathematics. It is used to identify the coordinates on a Cartesian plane, typically represented as \((x, y)\). The first element \(x\) signifies the horizontal position, and the second element \(y\) denotes the vertical position. Ordered pairs are essential for plotting points, graphing equations, and determining the location of points in two-dimensional spaces.

• The order in which elements appear in a pair is crucial; \((2, 5)\) is not the same as \((5, 2)\).
• Ordered pairs play a significant role when testing whether a specific point satisfies a given equation.

In the original exercise, the ordered pair \((2,5)\) refers to a point where \(x = 2\) and \(y = 5\). When determining whether this pair satisfies an equation, we substitute these values into the equation to see if the equation holds true.

Substitution Method

The substitution method is a powerful tool for solving equations and consists of replacing a variable with its corresponding value. This technique lets you test if a given point or solution fits into an equation, often for verifying the truth of a statement.
Here's how the substitution method was used in the original exercise:

• Identify the values for \(x\) and \(y\) from the ordered pair. In this case, \(x = 2\) and \(y = 5\).
• Insert these values into the equation \(3y - 2x = -4\). This creates a numerical expression instead of an algebraic one.

The result of substituting is the evaluation of the expression. In our example, substituting \(x = 2\) and \(y = 5\) into \(3y - 2x\), we simplify \(3(5) - 2(2)\) to get 15 - 4, which equals 11. Since 11 does not equal -4, the original statement is false.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and mathematical operations that represent a specific value or range of values. In the context of linear equations, they help describe relationships between variables. These expressions lay the groundwork for forming equations that can be used to solve real-world problems.
Consider the expression \(3y - 2x\). It's a linear expression involving operations between two variables \(x\) and \(y\). When exploring equations, such as \(3y - 2x = -4\), find values for one or both variables to verify if an ordered pair satisfies the equation.

• The key is simplifying expressions by combining like terms or substituting values to evaluate the statement's validity.
• Expressions can vary from simple linear forms to more complex polynomial equations depending on the problem.

Understanding and manipulating algebraic expressions are crucial for solving linear equations, letting you verify solutions as seen in the original exercise.