Problem 40 - Algebra A Combined Function

Question

Solve. See Examples 1 through 7 $$ -(4 a-7)-5 a=10+a $$

Step-by-Step Solution

Answer

\(a = -\frac{3}{10}\)

Step 1: Distribute the negative sign

The equation is \(-(4a - 7) - 5a = 10 + a\). First, distribute the negative sign inside the parenthesis: \[-(4a - 7) \rightarrow -4a + 7\]This transforms the equation into \(-4a + 7 - 5a = 10 + a\).

Step 2: Combine like terms

Now, combine like terms on the left side:\[-4a - 5a + 7 = -9a + 7\]This simplifies the equation to \(-9a + 7 = 10 + a\).

Step 3: Move variables to one side

Subtract \(a\) from both sides to get all terms containing \(a\) on one side:\[-9a + 7 - a = 10 + a - a\]This simplifies to \(-10a + 7 = 10\).

Step 4: Isolate the variable term

Subtract 7 from both sides to isolate the term with \(a\):\[-10a + 7 - 7 = 10 - 7\]This results in \(-10a = 3\).

Step 5: Solve for a

Divide both sides by \(-10\) to solve for \(a\):\[a = \frac{3}{-10}\]Thus, \(a = -\frac{3}{10}\).

Key Concepts

Distributive Property

The distributive property is a fundamental principle in algebra that allows you to simplify expressions and equations. It states that any term outside the parentheses can be distributed across the terms inside the parentheses. This property is described by the formula:

• If you have an equation like \(a(b+c)\), it can be expanded to \(ab + ac\).

In the original exercise, we see the distributive property at work as follows:

• The equation starts with \(- (4a - 7)\).
• Using the distributive property, the negative sign out front means you need to multiply \(-1\) by each term within the parenthesis, resulting in \(-4a + 7\).

This simplifies the equation and makes it easier to solve, as it removes the parentheses and focuses on each individual term.

Combining Like Terms

Combining like terms is a simple but powerful strategy used to simplify algebraic expressions and equations by merging terms with the same variables. Terms are considered 'like' if they have the same variable raised to the same power.

In the exercise, after distributing the negative sign, we have the equation:

• \(-4a + 7 - 5a = 10 + a\).

Here:

• The terms \(-4a\) and \(-5a\) are like terms because they both contain the variable \a\.
• By combining them, you get \(-9a\).

It's like gathering things that belong together, which is crucial in making the equation simpler and more manageable. In this step, the focus is only on the terms with the variable, not the constants or numbers.

Isolating Variables

Isolating variables is crucial in solving equations, as it aims to get the variable on one side of the equation while having everything else on the opposite side. In algebra, this means shifting all terms involving the variable you want to solve for to one side.

In the exercise:

• After combining like terms, the equation becomes: \(-9a + 7 = 10 + a\).
• To isolate the variable \a\, move \a\ to the left side by subtracting it from both sides, yielding \(-10a + 7 = 10\).

By getting all terms with \a\ on one side, the equation starts to become clearer. This process usually involves addition or subtraction, and it helps to simplify the solving process by narrowing down the focus to the variable of interest.

Solution of Equations

The solution of an equation completes the process of solving an algebraic equation, ensuring the unknown variable is now isolated and understood. In the final steps of solving an algebra problem, you aim to find the value of the variable.

For our exercise:

• The equation \(-10a = 3\) shows the variable is isolated, but not yet solved.
• To find what \a\ equals, divide both sides of the equation by \(-10\) which is currently multiplying \a\.
• This results in \(a = \frac{3}{-10}\), simplifying to \(a = -\frac{3}{10}\).

This approach ensures a clear and definite answer for \a\ is reached, marking the completion of the algebraic process. More importantly, checking the solution can confirm that all steps were done correctly.