Problem 29 - Algebra A Combined Function

Question

Solve each equation. Check each solution. See Examples 7 and 8 . \(\frac{2}{3} y-11=-9\)

Step-by-Step Solution

Answer

The solution is \( y = 3 \).

Step 1: Simplify the Equation

Start by isolating the variable on one side. The given equation is \( \frac{2}{3} y - 11 = -9 \). First, add 11 to both sides to get rid of the constant term on the left side:\[ \frac{2}{3} y - 11 + 11 = -9 + 11 \]This simplifies to:\[ \frac{2}{3} y = 2 \]

Step 2: Clear the Fraction

To eliminate the fraction, multiply both sides of the equation by the reciprocal of \( \frac{2}{3} \), which is \( \frac{3}{2} \). This will help isolate \( y \):\[ \frac{3}{2} \cdot \frac{2}{3} y = 2 \cdot \frac{3}{2} \]Simplifying, we get:\[ y = 3 \]

Step 3: Check the Solution

To ensure the solution is correct, substitute \( y = 3 \) back into the original equation:\[ \frac{2}{3} \cdot 3 - 11 = -9 \]Simplify the left side:\[ 2 - 11 = -9 \]Since both sides of the equation are equal, \( y = 3 \) is a correct solution.

Key Concepts

Isolating the Variable

When solving linear equations, the goal is to find the value of the variable that makes the equation true. This often means rearranging the equation so that the variable is by itself on one side. Here’s how you can approach isolating the variable:

• Identify the variable you need to isolate. In this exercise, the variable is \( y \).
• Begin by getting rid of any constants present with the variable. For our exercise, that means adding 11 to both sides of the equation \( \frac{2}{3} y - 11 = -9 \).
• Perform the arithmetic on both sides to simplify it. Doing this leaves you with \( \frac{2}{3} y = 2 \).

Remember, whatever operation you perform on one side of the equation, you must also do to the other. This rule keeps the equation balanced.

Eliminating Fractions

Fractions can make equations look complex, but eliminating them can simplify the situation significantly. Here’s a straightforward method:

• Identify the fraction involved with the variable. In this case, it's \( \frac{2}{3} \).
• Multiply both sides of the equation by the reciprocal of the fraction. The reciprocal of \( \frac{2}{3} \) is \( \frac{3}{2} \).
• Apply this multiplication to both sides: \( \frac{3}{2} \cdot \frac{2}{3} y = 2 \cdot \frac{3}{2} \).
• This step cancels out the fraction, leaving you with \( y = 3 \).

Multiplying by the reciprocal is a powerful tool for simplifying equations and reaching an isolated variable.

Checking Solutions

After calculating a solution, it’s essential to verify that it works in the original equation. Here's how to check your work:

• Take your calculated value of the variable, which in this example is \( y = 3 \).
• Substitute it back into the original equation: \( \frac{2}{3} \cdot 3 - 11 = -9 \).
• Simplify both sides of the equation. You should arrive at \( 2 - 11 = -9 \).
• If both sides are equal, your solution is correct! If not, reconsider your steps.

Checking your solution is crucial as it confirms your method was sound and that the calculated value is, indeed, correct. This practice builds confidence and accuracy in solving equations.