Problem 48 - Algebra for College Students

Question

Solve each equation. \(4(2 a+3)-3(4 a-2)=5(4 a-7)\)

Step-by-Step Solution

Answer

The solution for \(a\) is \( \frac{53}{24} \).

Step 1: Expand the Equation

First, expand all the terms in the equation by distributing the numbers outside the parentheses. Start with the left side:\[ 4(2a + 3) = 8a + 12 \]\[-3(4a - 2) = -12a + 6 \]Combine these results:\[ 8a + 12 - 12a + 6 \]Now, expand the right side:\[ 5(4a - 7) = 20a - 35 \]

Step 2: Simplify and Combine Like Terms

Simplify the left side of the equation by combining like terms:\[ 8a + 12 - 12a + 6 = -4a + 18 \]The entire equation now looks like this:\[ -4a + 18 = 20a - 35 \]

Step 3: Move Variable Terms to One Side

To isolate the variable terms, add \(4a\) to both sides of the equation:\[ -4a + 18 + 4a = 20a - 35 + 4a \]This simplifies to:\[ 18 = 24a - 35 \]

Step 4: Move Constant Terms to Other Side

Add 35 to both sides of the equation to get the constants on one side:\[ 18 + 35 = 24a \]This simplifies to:\[ 53 = 24a \]

Step 5: Solve for the Variable

To solve for \(a\), divide both sides by 24:\[ a = \frac{53}{24} \]

Step 6: Simplify the Solution

Since 53 and 24 have no common factors, the fraction cannot be simplified further. Therefore, the solution remains as:\[ a = \frac{53}{24} \]

Key Concepts

Distributive Property

The distributive property is a fundamental algebraic principle you will encounter often. It allows us to get rid of parentheses by multiplying out the numbers and variables inside them by the number outside. In the given equation, we applied the distributive property as follows:

• First, expand the expression on the left: \(4(2a+3)\) becomes \(8a+12\).
• Next, tackle \(-3(4a-2)\) on the left: This turns into \(-12a + 6\).
• On the right side, expand \(5(4a-7)\) to get \(20a-35\).

By successfully applying the distributive property, we transformed the equation into an expression without any parentheses, making it easier to work with.

Combining Like Terms

Once you have expanded the equation using the distributive property, it's time to combine like terms. Like terms are terms that have identical variables raised to the same power. Let's simplify the left side:

• Combine the terms \(8a\) and \(-12a\) to get \(-4a\).
• Then, add the constants \(12\) and \(6\) to obtain \(18\).

By combining these like terms, we condensed the equation to \(-4a + 18\) on the left, making it easier to focus on manipulating the variable.

Isolating the Variable

Isolating the variable is key to solving linear equations. This process involves moving all variable terms to one side and constant terms to the opposite side of the equation. Here's how we do it:

• From the equation \(-4a + 18 = 20a - 35\), we add \(4a\) to both sides to move all the variable terms to one side: \(18 = 24a - 35\).
• Then, add \(35\) to both sides to move constant terms to the other side, giving us \(53 = 24a\).

At this point, the variable \(a\) is isolated when it stands alone on one side, making it straightforward to solve for it.

Fraction Simplification

After isolating the variable, the next step involves solving for it, which might result in a fraction. Simplification is the final step:

• Divide both sides by \(24\) to solve for \(a\), leading to \(a = \frac{53}{24}\).
• Check to see if the fraction can be simplified. Since \(53\) and \(24\) share no common divisors, the fraction is already in its simplest form.

Understanding fraction simplification ensures that your answer is not only correct but also presented in the simplest way possible, easing further calculations or interpretations.