Problem 2 - Elementary Algebra

Question

For exercises 1-10, (a) solve. (b) check. $$ \frac{3}{5} x-\frac{1}{4}=\frac{9}{10} $$

Step-by-Step Solution

Answer

The solution to the equation is \( x = \frac{23}{12} \). Verification shows this solution is correct.

Step 1 - Eliminate the fractions

To make the equation easier to solve, eliminate the denominators 5, 4, and 10. The least common multiple (LCM) of these denominators is 20. Multiply every term by 20.\[ 20 \times \frac{3}{5} x - 20 \times \frac{1}{4} = 20 \times \frac{9}{10} \]This simplifies to: \[ 12x - 5 = 18 \]

Step 2 - Isolate the variable term

Add 5 to both sides of the equation to isolate the term containing the variable.\[ 12x - 5 + 5 = 18 + 5 \]This simplifies to: \[ 12x = 23 \]

Step 3 - Solve for the variable

Divide both sides by 12 to solve for x.\[ x = \frac{23}{12} \]

Step 4 - Check the solution (Part b)

Substitute \( x = \frac{23}{12} \) back into the original equation to verify the solution.Original equation:\[ \frac{3}{5} x - \frac{1}{4} = \frac{9}{10} \]Substitute \( x = \frac{23}{12} \):\[ \frac{3}{5} \left( \frac{23}{12} \right) - \frac{1}{4} \]Simplify the expression:\[ \frac{69}{60} - \frac{1}{4} \]Convert \( \frac{1}{4} \) to a fraction with denominator 60:\[ \frac{69}{60} - \frac{15}{60} = \frac{54}{60} = \frac{9}{10} \]This confirms the solution is correct.

Key Concepts

least common multiple

When solving linear equations with fractions, it helps to eliminate the fractions first. This is done by finding the least common multiple (LCM) of the denominators. The LCM is the smallest number that all denominators can divide into without leaving a remainder. For instance, consider the denominators 5, 4, and 10. The LCM of these numbers is 20. By multiplying each term in the equation by 20, we turn the fractions into whole numbers:\[ 20 \times \frac{3}{5} x - 20 \times \frac{1}{4} = 20 \times \frac{9}{10} \]Simplifying this gives:\[ 12x - 5 = 18 \]This step makes solving the equation much easier by removing the fractional coefficients.

isolate the variable

To solve the equation, we need to isolate the variable. This means we want the variable, like x, alone on one side of the equation. Starting with our simplified equation:\[ 12x - 5 = 18 \]We'll add 5 to both sides to remove the -5 on the left side:\[ 12x - 5 + 5 = 18 + 5 \]This simplifies to:\[ 12x = 23 \]Now, to get x by itself, divide both sides by 12:\[ x = \frac{23}{12} \]Now x is isolated, and we found that x equals \( \frac{23}{12} \).

checking solutions in algebra

It's crucial to check if the solution is correct by substituting it back into the original equation. This step ensures that no mistakes were made. Start with the original equation:\[ \frac{3}{5} x - \frac{1}{4} = \frac{9}{10} \]Substitute \( x = \frac{23}{12} \):\[ \frac{3}{5} \left( \frac{23}{12} \right) - \frac{1}{4} \]Simplify the multiplication first:\[ \frac{69}{60} - \frac{1}{4} \]Convert \( \frac{1}{4} \) to a fraction with a denominator of 60:\[ \frac{1}{4} = \frac{15}{60} \]Now perform the subtraction:\[ \frac{69}{60} - \frac{15}{60} = \frac{54}{60} \]Simplify \( \frac{54}{60} \) to \( \frac{9}{10} \), which matches the right side of the original equation. This confirms our solution \( x = \frac{23}{12} \) is correct.