Problem 8 - Prealgebra

Question

Solve each equation. Check your solution. $$3.5 a=7$$

Step-by-Step Solution

Answer

The solution is \(a = 2\).

Step 1: Understand the Equation

The equation we have is \(3.5a = 7\). This means that 3.5 times some number \(a\) equals 7. Our goal is to find the value of \(a\).

Step 2: Isolate the Variable

To solve for \(a\), we need to isolate it on one side of the equation. Since \(a\) is being multiplied by 3.5, we do the opposite operation to both sides of the equation, which is division.

Step 3: Divide by 3.5

Divide both sides of the equation by 3.5 to solve for \(a\): \[a = \frac{7}{3.5}\]When you do this calculation, \( \frac{7}{3.5} = 2\). So, \(a = 2\).

Step 4: Check the Solution

To ensure the solution is correct, substitute \(a = 2\) back into the original equation: \[3.5 \times 2 = 7\]Calculate the left side: \[7 = 7\]Both sides of the equation match, confirming that \(a = 2\) is the correct solution.

Key Concepts

Solving Equations

Equations are like puzzles where we need to find the missing piece or number. Solving equations means finding the value of the variable that keeps the equation balanced. In our problem, the equation given is \(3.5a = 7\). The variable we want to find here is \(a\). We solve equations by isolating the variable on one side to find its value.To isolate the variable \(a\), we look at what operations are being performed on it. If \(a\) is being multiplied by 3.5, the reverse operation to cancel this out is division. By dividing both sides of the equation by 3.5, we can isolate \(a\):

• Divide both sides: \(a = \frac{7}{3.5}\)
• Simplify to get: \(a = 2\)

This is how we find out that \(a\) must be 2 to balance the equation.

Algebra

Algebra involves using letters and symbols to represent numbers in equations and expressions. Here, we have used the letter \(a\) to stand for our unknown number. When engaging with algebra, we use rules of arithmetic and symbols to solve for unknowns.The equation \(3.5a = 7\) uses a basic algebraic principle. We applied the reverse operation of multiplication, which is division, to solve for \(a\). Knowing these basics equips us not only to solve equations like this but to tackle more complex algebraic expressions later on.Understanding algebra helps us:

• Organize and solve real-world problems.
• Develop logical thinking skills.
• Prepare for advanced math topics.

Check Solutions

Checking solutions is an important step to ensure your answer is correct. After solving an equation, we substitute the solution back into the original equation. This confirms that our solution is accurate.For our equation, we found \(a = 2\). To check this, substitute \(a = 2\) back into \(3.5a = 7\):

• Plug \(a = 2\) into the left side of the equation: \(3.5 \times 2\)
• Solve it: \(3.5 \times 2 = 7\)
• Check if it matches the right side: it does, as both sides equal 7

This shows that our solution \(a = 2\) is indeed correct. Always check your solutions to reduce errors and gain confidence in your answer.