Problem 33 - Introductory Algebra for College Students

Question

Solve each equation and check your proposed solution in Exercises. Begin your work by rewriting each equation without fractions. $$\frac{2 x}{3}-5=7$$

Step-by-Step Solution

Answer

The solution to the equation \(\frac{2 x}{3}-5=7\) is \(x = 18\).

Step 1: Eliminate Fractions

In order to eliminate the fractions, one can multiply the whole equation by 3, which will make it a bit simpler. The equation becomes: \(3*(\frac{2 x}{3}) - 3*5 = 3*7\), or simplified: \(2x - 15 = 21\).

Step 2: Isolate the Variable

Try to isolate x by adding 15 to both sides of the equation: \(2x - 15 + 15 = 21 + 15\), or simplified: \(2x = 36\).

Step 3: Solve for Variable

Finally, divide both sides of the equation by 2, to solve for x, as such: \(\frac{2x}{2} = \frac{36}{2}\), which gives: \(x = 18\).

Step 4: Check Solution

Check the solution by substituting \(x = 18\) back into the original equation, this would give: \(\frac{2 * 18}{3} - 5\), which simplifies to \(12 - 5 = 7\), so the solution is correct.

Key Concepts

Solving Equations

To solve algebraic equations, we follow systematic steps to find the value of the unknown, often represented by a variable like \(x\). The goal is to make sure both sides of the equation are balanced and equivalent.

Generally, here’s an approach you can adopt:

• First, simplify both sides of the equation as much as possible by performing any obvious arithmetic operations.
• Next, aim at making the variable stand alone on one side by "undoing" what is done to it through mathematical operations like addition, subtraction, multiplication, or division.
• Finally, verify your solution by substituting it back into the original equation. This ensures no mistakes were made during the process.

Recognizing the type of operations required and undoing them in reverse order takes practice, but it forms the foundation for solving any equation successfully.

Fractions in Equations

Fractions can often make solving equations seem tricky, but they can be managed easily by eliminating them. This is done by finding the denominator and multiplying the entire equation by it, which helps in handling the fraction components smoothly.

For instance, if you have an equation like \(\frac{2x}{3} - 5 = 7\), to eliminate the fraction, we multiply all terms in the equation by 3 (the denominator of the fraction). This leads to the new equation: \(2x - 15 = 21\).

• Multiplying an equation by the denominator makes the fraction disappear, simplifying the entire equation.
• Always multiply every term on both sides to maintain the balance of the equation.
• Once the fractions are gone, solving the equation becomes much more straightforward.

By routinely using this technique, equations containing fractions become easier to solve.

Variable Isolation

Isolating the variable is a fundamental skill in solving equations. It involves transforming the equation to get the variable alone on one side. This process signifies finding the value of the variable that makes the equation true.

In our example, once the fraction is removed, the equation is \(2x - 15 = 21\). To isolate \(x\), we start by reversing the subtraction with an addition: add 15 on both sides to get \(2x = 36\).

• Perform the same operation on both sides to maintain the equation's balance.
• Reverse the operation of multiplication by dividing each side by 2, resulting in \(x = 18\).
• The goal of this process is to have the variable, by itself, on one side of the equation.

The careful, step-by-step isolation of the variable ensures accuracy and correctness, paving the way for more complex algebraic manipulations.