Problem 38 - Algebra A Combined Function

Question

Solve. See Examples 1 through 7 $$ \frac{4(5-w)}{3}=-w $$

Step-by-Step Solution

Answer

The solution is \(w = 20\).

Step 1: Distribute the 4

Start by distributing the 4 inside the parentheses. This means you multiply 4 by each term inside the parentheses.\[ 4(5 - w) = 4 \times 5 - 4 \times w = 20 - 4w \]

Step 2: Rewrite the Equation

Substitute the distributed expression back into the equation. The equation now looks like this:\[ \frac{20 - 4w}{3} = -w \]

Step 3: Eliminate the Fraction

Multiply every term in the equation by 3 to eliminate the fraction. This gives:\[ 20 - 4w = -3w \]

Step 4: Collect Like Terms

Add \(4w\) to both sides of the equation to move all the terms involving \(w\) to one side.\[ 20 = 4w - 3w \]

Step 5: Simplify

Simplify the equation:\[ 20 = w \]

Step 6: Verify the Solution

Plug \(w = 20\) back into the original equation to ensure it is correct:\[ \frac{4(5 - 20)}{3} = -20 \]\[ \frac{4(-15)}{3} = -20 \]\[ \frac{-60}{3} = -20 \]\[ -20 = -20 \]The solution is correct.

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps to simplify expressions and solve equations. In simple terms, it allows you to multiply a single term by each term within a set of parentheses. For example, if you have a term like \(4(5-w)\), applying the distributive property means you multiply 4 by both 5 and \(-w\).

This results in:

• \(4 \times 5 = 20\)
• \(4 \times (-w) = -4w\)

So, the expression \(4(5-w)\) simplifies to \(20 - 4w\). By distributing at the start of solving an equation, you reduce the complexity of further operations needed to isolate variables.

Using the distributive property correctly is crucial for breaking down more complex expressions into manageable pieces.

Fractions in Equations

When you are working with equations that include fractions, it's typically easier to solve them by eliminating the fractions early on. This simplifies the process and reduces potential errors.

In the expression \(\frac{20 - 4w}{3} = -w\), the fraction can be eliminated by multiplying every term by the denominator, which is 3 in this case. This operation serves to clear the fraction for a clearer path to solving for \(w\).

Let's see the steps:

• Multiply both sides of the equation by 3, aiming to eliminate the fraction: \(3 \times \frac{20-4w}{3} = 3 \times (-w)\)
• This simplifies to: \(20 - 4w = -3w\).

By removing the fraction, you transform the equation into a linear one, which is easier to handle.

Like Terms

In algebra, identifying and combining like terms is a key skill for simplifying expressions and solving equations. Like terms are terms that have the same variable raised to the same power. For instance, in the equation \(20 - 4w = -3w\), \(-4w\) and \(-3w\) are like terms because they both include the variable \(w\).

The process of combining like terms in this context involves moving all similar terms to one side of the equation to simplify it.

Here's how it works:

• Add \(4w\) to both sides: \(20 = -3w + 4w\)
• This simplifies to: \(20 = w\)

Gathering like terms on one side often simplifies the equation and helps to solve for the unknown variable.

Algebraic Verification

Verifying your solution is an essential step to ensure accuracy in solving algebraic equations. It involves substituting the found solution back into the original equation to see if it holds true.

For instance, with the solution \(w = 20\) from the equation \(\frac{4(5-w)}{3} = -w\), verify by substituting \(w = 20\) back:

• Original equation: \(\frac{4(5 - 20)}{3} = -20\)
• Simplifies to: \(\frac{4(-15)}{3} = -20\)
• Which further simplifies to: \(\frac{-60}{3} = -20\)
• And confirms: \(-20 = -20\)

When both sides of the equation are equal, the solution is verified. This step confirms the accuracy of your solution and ensures no mistakes were made in the solving process.