Problem 43 - Elementary Algebra

Question

Solve the equations. $$ -7(2 a-1)=63 $$

Step-by-Step Solution

Answer

Answer: a = -4

Step 1: Distribute the -7 to both terms within the parentheses

To begin, we will remove the parentheses by applying the distributive property, which means we will multiply -7 by both terms inside the parentheses. Here's the result: $$ -7(2 a-1) = -14a + 7. $$ Now our equation looks like this: $$ -14a + 7 = 63. $$

Step 2: Subtract 7 from both sides of the equation

To isolate the "a" term, we will first move the constant term (+7) to the other side of the equation by subtracting 7 from both sides: $$ -14a + 7 - 7 = 63 - 7. $$ After simplification, we obtain the following equation: $$ -14a = 56. $$

Step 3: Divide by -14 to solve for 'a'

Now, we need to isolate "a" by dividing both sides of the equation by -14, the coefficient of 'a': $$ \frac{-14a}{-14} = \frac{56}{-14}. $$ Upon performing the division, we get the value of 'a': $$ a = -4. $$

Step 4: Check the solution

To make sure our solution is correct, we can substitute the value of 'a' back into the original equation and see if it holds true: $$ -7(2(-4) - 1) = 63. $$ Calculating the left-hand side of the equation gives us: $$ -7(-8 - 1) = -7(-9) = 63. $$ Since the left-hand side equals the right-hand side, our solution of a = -4 is correct.

Key Concepts

Understanding the Distributive Property

The distributive property is a key tool in algebra that helps us to simplify and solve equations. When you come across an expression that involves a multiplication of a number by a sum or a difference inside parentheses, you're likely dealing with the distributive property. The goal here is to "distribute" the multiplying factor across the terms inside the parentheses.
For example, in the expression \(-7(2a - 1)\), we distribute \(-7\) by multiplying it with each term inside the parentheses. This means:

• Multiply \(-7\) by \(2a\) which gives us \(-14a\).
• Next, multiply \(-7\) by \(-1\) which gives us \(7\).

Thus, the equation \(-7(2a - 1)\) becomes \(-14a + 7\). This step is crucial for simplifying the equation and setting the stage for isolating the variable.

Isolating the Variable

Once you've applied the distributive property, the next step often involves isolating the variable—this means getting the variable alone on one side of the equation. This process requires you to peel back layers of numbers and operations until the variable stands by itself.
To illustrate, in the equation \(-14a + 7 = 63\), the variable \(a\) is trapped with a \(-14\) coefficient and a \(+7\) constant. Here's how to isolate \(a\):

• First, eliminate the constant by subtracting \(7\) from both sides. This gives us \(-14a = 56\).
• Second, remove the coefficient \(-14\) by dividing each side by \(-14\). This leaves \(a\) by itself: \(a = -4\).

By patiently and strategically isolating the variable, we can easily determine its value.

Checking Solutions for Accuracy

Solving an equation doesn't end with finding a potential solution. Checking your work is vital to confirm that the solution is indeed correct. This step involves plugging the found value back into the original equation to see if both sides equal.
For instance, if we substitute \(a = -4\) into the original equation \(-7(2a - 1) = 63\), the process would look like this:

• Replace \(a\) with \(-4\), giving us: \(-7(2(-4) - 1)\).
• Simplify the inside of the parentheses: \(2(-4) - 1 = -9\).
• Multiply: \(-7)(-9) = 63\).

Both sides equate to 63, confirming that our solution \(a = -4\) is accurate. Checking solutions ensures errors are caught and provides confidence in the results.