Problem 66 - Algebra 2

Question

Determine whether \(\frac{2 d+5}{3 d+5}=\frac{2}{3}\) is sometimes, always, or never true. Explain.

Step-by-Step Solution

Answer

The equation is never true.

Step 1: Cross Multiply

To determine if the equation \(\frac{2d+5}{3d+5}=\frac{2}{3}\) is sometimes, always, or never true, let's start by cross-multiplying both sides of the equation. This gives us: \[3(2d+5) = 2(3d+5)\].

Step 2: Distribute Terms

Next, we distribute the constants on both sides of the equation. Distributing 3 gives \(6d + 15\), and distributing 2 gives \(6d + 10\). This results in the equation: \[6d + 15 = 6d + 10\].

Step 3: Simplify the Equation

Subtract \(6d\) from both sides of the equation to eliminate the \(d\) terms. This simplification gives \[15 = 10\].

Step 4: Analyze Result

The simplified equation \(15 = 10\) is clearly false. This means there are no values of \(d\) that make the original equation true. The equation \(\frac{2d+5}{3d+5}=\frac{2}{3}\) is never true.

Key Concepts

Cross Multiplication

Cross multiplication is a technique used to make solving equations involving fractions much more manageable. It is particularly handy when dealing with equations of the form \( \frac{a}{b} = \frac{c}{d} \). The principle behind cross multiplication is to eliminate the fractions by multiplying across the equality sign. For instance, in the equation \( \frac{2d+5}{3d+5} = \frac{2}{3} \), cross-multiplying involves taking the numerator of one side and multiplying it by the denominator of the other side:

• Multiply \(3\) by \(2d+5\) to get \(3(2d+5)\).
• Similarly, multiply \(2\) by \(3d+5\) to get \(2(3d+5)\).

This gives us the new equation \(3(2d+5) = 2(3d+5)\), which is easier to manage since it no longer has fractions. This technique preserves the equality because it effectively multiplies both sides by the product of the two original denominators, simplifying the equation-solving process.

Simplification

Simplification is the process of reducing an equation or expression to its simplest form. This often involves combining like terms, canceling out terms, or performing arithmetic operations. In our original problem, we simplify the equation after cross multiplication. We have derived \(3(2d+5) = 2(3d+5)\) and need to simplify further. Let's follow:

• After distributing, we get \(6d + 15 = 6d + 10\).
• Here, we can subtract \(6d\) from both sides to eliminate the \(d\) terms. After doing this, we end up with \(15 = 10\).

This simplification process is key as it led us to realize that the final equation is incorrect (since \(15 = 10\) is not a true statement). Therefore, it indicates that no value for \(d\) satisfies the initial equation, concluding that it is never true.

Distributive Property

The distributive property is a valuable algebraic tool that allows you to multiply a sum by distributing the multiplication to each addend individually. It's often used in solving equations to eliminate parentheses and simplify expressions. This property states that \(a(b + c) = ab + ac\). We use this property when expanding the equation from our problem. For instance:

• \(3(2d+5)\) is distributed as \(3 \times 2d + 3 \times 5\), resulting in \(6d + 15\).
• \(2(3d+5)\) is distributed as \(2 \times 3d + 2 \times 5\), giving \(6d + 10\).

Using the distributive property, we simplify the equation from our cross-multiplication step, getting rid of the parentheses and revealing a simpler form of the equation to work with. This step is crucial for making further simplification possible and moving towards finding if any solutions exist.

Algebraic Equations

Algebraic equations are mathematical statements that involve variables and constants. They represent relationships where one side of the equation is set equal to the other side. Solving algebraic equations involves figuring out the value(s) of the variable(s) that make the equation true.In the context of the original exercise, we are dealing with a rational algebraic equation \( \frac{2d+5}{3d+5} = \frac{2}{3}\). The goal is to determine if any values of \(d\) satisfy this equation. The steps to solve algebraic equations usually involve:

• Using methods like cross multiplication to remove fractions.
• Simplifying the equation using properties like distributive property.
• Carrying out further simplifications to isolate the variable.
• Analyzing the final result to confirm the existence of solutions.

In our solved example, the absence of a consistent value for \(d\) that satisfies the equation illustrates a crucial lesson in algebra: sometimes, equations can have no solution. Understanding this helps in correctly interpreting and solving algebraic equations.