Problem 43 - Fundamentals of Mathematics

Question

Solve each equation. Be sure to check each result. $$ \frac{5 a}{7}=10 $$

Step-by-Step Solution

Answer

The solution is \( a = 14 \).

Step 1: Understand the Equation

The equation given is \( \frac{5a}{7} = 10 \). We need to solve for \(a\).

Step 2: Clear the Fraction

Multiply both sides of the equation by 7 to clear the fraction: \[ 7 \times \frac{5a}{7} = 10 \times 7 \]. This simplifies to \(5a = 70\).

Step 3: Isolate the Variable

Divide both sides by 5 to solve for \(a\): \[ a = \frac{70}{5} \]. This simplifies to \(a = 14\).

Step 4: Check the Solution

Substitute \(a = 14\) back into the original equation to verify: \( \frac{5 \times 14}{7} = 10 \). Simplifying, \( \frac{70}{7} = 10\), which equals 10, confirming the solution is correct.

Key Concepts

Checking Solutions

When solving equations, checking your solution is a crucial step. After finding what you think is the answer, you should always return to the original equation to verify that your solution is correct. This involves substituting the value you've found back into the original equation and ensuring that both sides are equal.
Why is this important?

• Ensures there's no mistake in calculations.
• Confirms the logical reasoning of each step.
• Provides confidence in the correctness of the solution.

In our case, after calculating that \(a = 14\), we substitute it back into \(\frac{5a}{7} = 10\). Plugging \(a = 14\) gives us \(\frac{5 \times 14}{7} = 10\), simplifying to \(10 = 10\). This tells us that our solution for \(a\) is indeed correct.

Fractions in Equations

Working with fractions in equations can seem daunting, but it's quite manageable with a straightforward approach. The main strategy is to clear the fraction by multiplying every term by the denominator. This effectively eliminates the fraction, simplifying the equation.
Why do we eliminate fractions?

• Fractions can make calculations more complicated.
• Clearing them helps to simplify the equation, making it easier to work with.
• Allows focus on isolating the variable without dealing with fractional coefficients.

In our example, we had the equation \(\frac{5a}{7} = 10\). By multiplying each side by 7, the fraction is cleared, changing the equation to \(5a = 70\). From this, it's much easier to proceed with isolating the variable \(a\).

Isolating Variables

Isolating variables is the process of getting the variable by itself on one side of the equation. The main goal is to solve for the variable, giving us its value. This is typically achieved through inverse operations—doing the opposite of what's currently done to the variable.
Steps to Isolate Variables:

• Identify the operations being performed on the variable.
• Apply inverse operations to each term to undo these operations.
• Continue until the variable is alone on one side of the equation.

Inserting this into our example after clearing fractions, we work with \(5a = 70\). The variable \(a\) is being multiplied by 5, so we use division (the inverse operation) by 5 to isolate \(a\). Thus, \(a = \frac{70}{5}\), leading to \(a = 14\). With \(a\) isolated, we've successfully solved the equation.