Problem 21 - Elementary and Intermediate Algebra

Question

Solve each equation. Check the result. $$ 7(a+2)=11 a+17-7 a $$

Step-by-Step Solution

Answer

The solution to the equation is \(a = 1\).

Step 1: Distribute on the Left Side

Expand the left side of the equation by distributing 7 across the terms in the parentheses: \[ 7(a + 2) = 7a + 14 \] This gives the equation: \[ 7a + 14 = 11a + 17 - 7a \]

Step 2: Simplify the Right Side

Combine like terms on the right side of the equation:\[ 11a - 7a + 17 = 4a + 17 \]So the equation becomes:\[ 7a + 14 = 4a + 17 \]

Step 3: Move Variables to One Side

Subtract \(4a\) from both sides to get all variable terms on one side:\[ 7a - 4a + 14 = 17 \]Simplifying gives:\[ 3a + 14 = 17 \]

Step 4: Solve for the Variable

Subtract 14 from both sides to isolate the term with \(a\):\[ 3a = 17 - 14 \]This simplifies to:\[ 3a = 3 \]Now, divide both sides by 3:\[ a = 1 \]

Step 5: Check the Solution

Substitute \(a = 1\) back into the original equation to verify:\[ 7(1 + 2) = 11(1) + 17 - 7(1) \]Calculate each side:Left side: \[ 7(3) = 21 \]Right side: \[ 11 + 17 - 7 = 21 \]Both sides are equal, confirming that \(a = 1\) is the correct solution.

Key Concepts

Distributive Property

The Distributive Property is a crucial rule in algebra that allows us to simplify expressions by dealing with parentheses. When dealing with an equation such as \(7(a+2)\), we use the distributive property to multiply 7 by each term inside the parentheses separately. Here, we distribute the 7 to both \(a\) and 2:

• Multiply 7 by \(a\): \(7 \times a = 7a\)
• Multiply 7 by 2: \(7 \times 2 = 14\)

This gives us the expanded expression: \(7a + 14\). This step is critical for transforming complex equations into simpler ones that are easier to work with.

Combining Like Terms

Combining like terms involves grouping similar terms to simplify an equation further. Essentially, like terms are terms that have the same variable raised to the same power. For instance, in the equation \(11a - 7a + 17\), the terms \(11a\) and \(-7a\) are like terms because both contain the variable \(a\). By combining these terms:

• \(11a - 7a = 4a\)
• This simplifies the expression to \(4a + 17\)

This step is important because it helps condense the equation, making it easier to isolate variables later in the solving process.

Solving Linear Equations

Solving linear equations involves manipulating the equation to find the value of the variable. Starting with a simplified equation, such as \(7a + 14 = 4a + 17\), you aim to isolate the variable by strategically using operations like addition or subtraction. Let's move all variable terms to one side by subtracting \(4a\) from both sides:

• \(7a - 4a = 3a\)

This changes the equation to \(3a + 14 = 17\). From this point, you simplify further until the equation becomes manageable enough to solve for the variable straightforwardly.

Variable Isolation

Variable isolation is the ultimate goal in solving an equation. This means getting the variable by itself on one side of the equation to determine its value. From the equation \(3a + 14 = 17\), our task is to isolate \(a\). Subtract 14 from both sides to remove the constant term from the left side:

• \(3a = 17 - 14\)
• This simplifies to \(3a = 3\)

Finally, divide both sides by 3 to solve for \(a\):

• \(a = 1\)

This process is essential because it provides the solution to the equation, allowing us to verify if our solution satisfies the original equation by substitution.