Problem 70 - Intermediate Algebra

Question

Solve each equation. $$ 7 a+2=12-4(a-3) $$

Step-by-Step Solution

Answer

The solution is \(a = 2\).

Step 1: Distribute on the Right Side

First, distribute the \, \(-4\) \, to both terms inside the parenthesis on the right-hand side of the equation \, \(12 - 4(a-3)\). This means multiplying \,\(-4\)\, by \,\(a\)\, and \,\(-4\)\, by \,\(-3\). The expression becomes \[12 - 4a + 12\].

Step 2: Simplify the Right Side

Combine like terms on the right side of the equation. By combining the constants \,\(12\)\, and \,\(12\), we get \[24 - 4a\].So now, our equation is \[7a + 2 = 24 - 4a\].

Step 3: Move all terms involving 'a' to one side

Add \,\(4a\)\, to both sides of the equation to move all terms involving \,\(a\)\, to the left side. This results in \[7a + 4a + 2 = 24\].Combine like terms to get \[11a + 2 = 24\].

Step 4: Isolate the 'a' term

Subtract \,\(2\)\, from both sides of the equation to isolate the \,\(a\)\, term. This gives us \[11a = 22\].

Step 5: Solve for 'a'

Finally, divide both sides by \,\(11\)\, to solve for \,\(a\). This yields \[a = 2\].

Key Concepts

Distributive Property

The distributive property is a key principle used in algebra to simplify expressions and solve equations. It allows you to multiply a single term by each term within a parenthesis. This property helps in breaking down complex expressions into simpler parts.
For instance, in the exercise provided, the application of the distributive property occurs when multiplying \(-4\) by both \(a\) and \(-3\) inside the parenthesis. Mathematically, this is represented as:

• \(-4 \cdot a = -4a\)
• \(-4 \cdot -3 = 12\)

After applying this property, the equation transforms into a more manageable form, which is crucial for the subsequent steps of solving the equation.

Combining Like Terms

Combining like terms helps to condense expressions by grouping similar terms together. This step simplifies the equation and sets the stage for isolating variables.
In our case, on the right side of the equation \(12 - 4a + 12\), you combine the constant terms \(12\) and \(12\) to get \(24\).
When simplifying, look for terms with:

• the same variable raised to the same power
• constant terms, which are numbers without variables

Combining like terms reduces complexity and helps clarify which terms need further manipulation.

Isolating Variables

Isolating the variable is a crucial step in solving linear equations. The goal is to get the variable by itself on one side of the equation. This makes it easier to determine the value of the variable.
In this example, isolating the variable \(a\) involves collecting all terms with \(a\) on one side and moving constants to the opposite side. To achieve this, we add \(4a\) to both sides, resulting in:

• \(7a + 4a + 2 = 24\)

Once like terms are combined, the equation becomes \(11a + 2 = 24\). The next step is subtracting \(2\) from both sides, leading to \(11a = 22\).
This prepares the equation for the final solving step.

Solving Equations

Solving equations involves finding the value of the variable that satisfies the equation. After isolating the variable, the equation \(11a = 22\) becomes straightforward to solve.
The final step is to divide both sides by \(11\) to find the value of \(a\):

• \(a = \frac{22}{11}\)

This simplifies to \(a = 2\).
In summary, solving for the variable after isolating it requires basic operations such as addition, subtraction, multiplication, or division, depending on the form of the equation. Understanding these operations ensures you can solve similar equations efficiently.