Problem 7 - Algebra A Combined Function

Question

Solve each equation. Check each solution. See Examples 1 through \(6 .\) \(\frac{2}{3} x=-8\)

Step-by-Step Solution

Answer

The solution is \(x = -12\).

Step 1: Isolate the variable

To solve the equation \(\frac{2}{3}x = -8\), we first need to isolate the variable \(x\). To get \(x\) alone, we multiply both sides of the equation by the reciprocal of \(\frac{2}{3}\), which is \(\frac{3}{2}\). This yields: \(x = -8 \times \frac{3}{2}\).

Step 2: Simplify the equation

Simplify the right side of the equation: \(x = -12\). We multiply \(-8\) by \(\frac{3}{2}\), which results in \(x = -12\).

Step 3: Check the solution

Substitute \(x = -12\) back into the original equation to verify the solution: \(\frac{2}{3} \times (-12)\). This calculation yields \(-8\), which is the right side of the original equation, confirming that \(x = -12\) is correct.

Key Concepts

Isolation of Variables

In mathematics, solving equations often involves a fundamental concept known as the isolation of variables. This means rearranging the equation so that the variable you're solving for appears alone on one side of the equation. To isolate the variable in linear equations, one must eliminate other terms or coefficients that are paired with the variable.

For example, in the equation \(\frac{2}{3}x = -8\), the goal is to isolate \(x\). Since \(x\) is multiplied by \(\frac{2}{3}\), we can get rid of this fraction by multiplying both sides of the equation by its reciprocal, which is \(\frac{3}{2}\). This operation will cancel out \(\frac{2}{3}\) on the left side, leaving \(x\) by itself.

Here are key steps for isolation of variables:

• Identify the term with the variable.
• Use inverse operations to move other terms to the opposite side of the equation.
• Keep the equation balanced by performing the same operation on both sides.

These steps help simplify the equation, making it possible to clearly see the value of the variable.

Checking Solutions in Algebra

After finding a potential solution to an equation, it’s essential to check that the solution satisfies the original equation. This confirmation step helps ensure no errors were made during the solving process.

To check the solution for the equation \(\frac{2}{3}x = -8\), after calculating \(x = -12\), you substitute \(-12\) back into the original equation to see if both sides are equal.

Substituting gives: \(\frac{2}{3} \times (-12)\). When you calculate this, the result is \(-8\), which matches the right side of the original equation. Thus, making sure your solution is indeed correct.

Important steps in checking solutions include:

• Substitute the solution back into the original equation.
• Simplify each side of the equation.
• Confirm that both sides of the equation are equal, verifying the correctness of the solution.

Always take the time to check your work—it builds accuracy and confidence.

Multiplying by Reciprocals

Multiplying by reciprocals is a handy technique used to cancel out fractions and solve equations that include fractions. In essence, the reciprocal of a number is what you multiply it by to get 1.

For instance, consider the fraction \(\frac{2}{3}\). Its reciprocal is \(\frac{3}{2}\) because \(\frac{2}{3} \times \frac{3}{2} = 1\). When solving equations like \(\frac{2}{3}x = -8\), you utilize this property when multiplying \(\frac{2}{3}\) by its reciprocal \(\frac{3}{2}\) to effectively "cancel" the fraction and simply have \(x\) on one side.

Key steps for multiplying by reciprocals include:

• Identify the fraction coefficient of the variable.
• Determine its reciprocal and multiply both sides of the equation by that reciprocal.
• This operation will cancel the fraction, isolating the variable.

Mastering this technique simplifies solving equations and eliminates the complexity that fractions can introduce.