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Chapter 2: Problem 29

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

Short Answer

Expert verified
The solution is \( y = 3 \).

Step by step solution

01

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:\[ \frac{2}{3} y = 2 \]
02

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:\[ y = 3 \]
03

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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鈥檚 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鈥檚 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鈥檚 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.

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Most popular questions from this chapter

The sum of the measures of the angles of a parallelogram is \(360^{\circ} .\) In the parallelogram below, angles \(A\) and \(D\) have the same measure as well as angles \(C\) and \(B\). If the measure of angle \(C\) is twice the measure of angle \(A,\) find the measure of each angle.

A quadrilateral is a polygon with 4 sides. The sum of the measures of the 4 angles in a quadrilateral is \(360^{\circ} .\) If the measures of the angles of a quadrilateral are consecutive odd integers, find the measures.

Solve each inequality. Write each answer using solution set notation. $$ 4-x<8 x+2 x $$

The room numbers of two adjacent classrooms are two consecutive even numbers. If their sum is 654 , find the classroom numbers.

If the sum of a number and five is tripled, the result is one less than twice the number. Find the number.
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