Problem 32 - Prealgebra

Question

Solve each equation. Check your solution. $$\frac{3}{4} a+4=\frac{1}{4}(3 a+16)$$

Step-by-Step Solution

Answer

The equation is an identity; it holds true for any value of \(a\).

Step 1: Eliminate Fractions

First, we should eliminate fractions by multiplying each term in the equation by 4 to get rid of the denominators. This gives us:\[4 \times \left( \frac{3}{4}a + 4 \right) = 4 \times \frac{1}{4}(3a + 16)\]Simplifying each side, we have:\[3a + 16 = 3a + 16\]

Step 2: Simplify and Solve for 'a'

At this point, both sides of the equation are identical (i.e., \(3a + 16 = 3a + 16\)), which indicates that the equation is an identity. This means that any value for \( a \) will satisfy the equation.

Step 3: Check Solution

To confirm, substituting any value for \(a\) should satisfy the equation. For example, let \(a = 0\):Substituting in the original equation, we have:\[\frac{3}{4}(0) + 4 = \frac{1}{4}(3 \cdot 0 + 16)\]This simplifies to:\[4 = \frac{16}{4} = 4\]This verifies the equation holds true. Since the original simplified equation was an identity, all values of \(a\) satisfy it.

Key Concepts

Identity Equation

An identity equation is a unique type of equation where both sides are equal for all possible values of the variable. This happens because the expressions on both sides simplify to the same expression. In our example, after simplifying the equation, we ended up with \[3a + 16 = 3a + 16\].
This indicates that no matter what value you substitute for \(a\), the equation will always hold true.
In practical terms, this means there are infinitely many solutions. While solving equations, an identity sometimes emerges after simplification, showing that the variables cancel out and the constants are equal. This is crucial in understanding that the solution encompasses all real numbers specific to the variable in question.

Solving Equations

Solving equations involves finding the value of the variable that makes the equation true.
To solve any equation, you generally want to isolate the variable on one side. This can sometimes lead us to special cases like identity equations. In step-by-step methods, you may encounter:

• Moving terms around by adding or subtracting.
• Using inverse operations like multiplication and division.
• Combining like terms and simplifying both sides.

These processes help in balancing the equation and getting closer to finding the solution.
In our problem, simplifying led us directly to an identity equation, emphasizing the value of simplification in solving.

Elimination of Fractions

Eliminating fractions is often the first step in solving equations with fractional coefficients.
Fractions can complicate calculations, so removing them can simplify the process significantly.
In our exercise: \[4 \times \left( \frac{3}{4}a + 4 \right) = 4 \times \frac{1}{4}(3a + 16)\]The aim was to multiply every term by 4, the least common denominator, canceling out the fractions.
This resulted in simplifying the expression to integers, making it easier to handle and solve:\[3a + 16 = 3a + 16\]Mastering this technique is valuable for handling any equation with fractions without being bothered by complex fractional arithmetic.

Check Solution

Checking the solution verifies if the value obtained satisfies the original equation. For identity equations, this step confirms the infinite validity of all numbers as solutions.
In our task, substituting any value, like \(a = 0\): \[\frac{3}{4}(0) + 4 = \frac{1}{4}(3 \cdot 0 + 16)\]illustrated that both sides simplified to 4, proving consistency:\[4 = \frac{16}{4} = 4\]Checking is straightforward but essential, as it uncovers any errors in calculation or reasoning along the way.
For identity equations, this process nicely highlights how any real number can satisfy the equality, affirming the identity.”