Problem 49 - Intermediate Algebra

Question

Solve each equation. Check each result. See Example 5. $$ 2(a-5)-(3 a+1)=0 $$

Step-by-Step Solution

Answer

The solution is \(a = -11\).

Step 1: Distribute the Terms

First, distribute the 2 across the terms inside the parentheses in the expression \(2(a - 5)\). This gives you: \[2a - 10\] So, the equation becomes: \(2a - 10 - (3a + 1) = 0\).

Step 2: Remove Parentheses

Apply the negative sign to both terms inside the parentheses \(-(3a + 1)\) resulting in \(-3a - 1\). Then the equation is:\[2a - 10 - 3a - 1 = 0\].

Step 3: Combine Like Terms

Combine the \(a\) terms and constant terms on the left side of the equation:\((2a - 3a) + (-10 - 1) = -a - 11\). The equation becomes:\[-a - 11 = 0\].

Step 4: Solve for \(a\)

To solve for \(a\), add 11 to both sides of the equation to isolate the term with \(a\):\[-a = 11\].Then, multiply or divide every term by -1 to solve for \(a\):\[a = -11\].

Step 5: Check the Solution

To verify the solution, substitute \(a = -11\) back into the original equation and see if it simplifies to a true statement:\[2(-11 - 5) - (3(-11) + 1) = 0\].Simplifying gives:\[2(-16) - (-33 + 1) = -32 - (-32) = 0\],which is a true statement. Thus, \(a = -11\) is correct.

Key Concepts

Understanding the Distributive Property

The distributive property is essential in solving linear equations. It allows you to simplify expressions by distributing a multiplier across terms in a parenthesis. Given our exercise, the expression \(2(a - 5)\) can be expanded using the distributive property. Here, you multiply the number outside the parenthesis (which is 2) by each term inside the parenthesis. Hence, you end up with:

• 2 multiplied by \(a\), resulting in \(2a\)
• 2 multiplied by \(-5\), yielding \(-10\)

Thus, \(2(a - 5)\) becomes \(2a - 10\). By applying this property, you simplify complex expressions and pave the way for easier solving of equations. Remember, mastering this property is key to handling more challenging equations!

Combining Like Terms

Once the distributive property is applied, the next step is combining like terms. Like terms have identical variable parts; for instance, in our example, 2a and -3a are like terms because they both involve the variable a.To combine these terms, you merely need to perform the arithmetic operation:

• Combine \(2a\) and \(-3a\) to get \(-a\)

Similarly, constant terms like -10 and -1 are also combined:

• Add \(-10\) to \(-1\), resulting in \(-11\)

Finally, combining gives you the equation \(-a - 11 = 0\). Simplifying equations by combining like terms helps in systematically reducing equations to simpler forms, making them easier to solve.

Checking Solutions in Equations

Checking your solution is a fundamental step to ensure its correctness. Once you have found the solution, substitute it back into the original equation to verify it results in a true statement. Let's see what this looks like with \(a = -11\) in our case.Insert \(-11\) in place of \(a\) in the original equation:

• \(2(-11 - 5)\)
• \(- (3(-11) + 1)\)

Simplify these expressions:

• \(2(-16) = -32\)
• \(- ( -33 + 1) = - (-32) = 32\)

Finally, because \( -32 + 32 = 0 \), the equation accurately balances. Confirming the solution ensures you understand each step in solving equations and assures accuracy, a crucial habit in mathematical practice.