Problem 81 - Algebra 2

Question

Solve each equation by using the method of your choice. Find exact solutions. \(9 y^{2}=49\)

Step-by-Step Solution

Answer

The solutions to the equation are \(y = \frac{7}{3}\) and \(y = -\frac{7}{3}\).

Step 1: Isolate the Quadratic Term

To solve the equation \(9y^2 = 49\), the first step is to isolate the term \(y^2\). We can do this by dividing both sides of the equation by 9: \[ y^2 = \frac{49}{9} \]

Step 2: Taking the Square Root

Now that we have \(y^2 = \frac{49}{9}\), take the square root of both sides to solve for \(y\). Remember to include both the positive and negative roots: \[ y = \pm \sqrt{\frac{49}{9}} \] Taking the square root, we have: \[ y = \pm \frac{\sqrt{49}}{\sqrt{9}} \]

Step 3: Simplify the Expression

Simplify the expression obtained from the square root calculation. Compute \(\sqrt{49}\) and \(\sqrt{9}\):\[ y = \pm \frac{7}{3} \]Thus, the exact solutions are \(y = \frac{7}{3}\) and \(y = -\frac{7}{3}\).

Key Concepts

Isolating Variables

When solving quadratic equations like \(9y^2 = 49\), the first important step is isolating the variable term, which in this case is \(y^2\). This means you want to get \(y^2\) all by itself on one side of the equation. This often involves some simple algebraic operations such as addition, subtraction, multiplication, or division.
In this example, we achieved isolation by dividing both sides by 9. By doing so, the equation becomes \(y^2 = \frac{49}{9}\). This step is crucial as it simplifies our equation and makes it ready for further solving.
Remember that the main goal here is to restructure the equation so that you have the quadratic term or the variable squared alone. This sets the stage for the next step, which involves using another operation to 'undo' the squaring.

Square Roots

After isolating \(y^2\), the next step in solving the quadratic equation is to take the square root of both sides. Taking a square root is the inverse operation of squaring, and it helps to find the value of the variable, \(y\), in our equation.
When you take the square root of a variable term that was squared, you must consider both positive and negative roots. This is because both \(7^2 = 49\) and \((-7)^2 = 49\). Hence, when we solve \(y^2 = \frac{49}{9}\) by taking the square root, we need to write \(y = \pm \sqrt{\frac{49}{9}}\).

• We apply the property that \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\), which helps us break down the expression into \(y = \pm \frac{\sqrt{49}}{\sqrt{9}}\).
• Finding square roots of constants simplifies the expression further.

Understanding this concept allows you to solve the equation and to see that taking square roots is a critical stage in determining the values of \(y\).

Exact Solutions

Achieving the exact solutions is the last part of solving the quadratic equation once the variable has been isolated and the square roots calculated. In our example, \(y = \pm \frac{7}{3}\). These values come from simplifying the square roots \(\sqrt{49} = 7\) and \(\sqrt{9} = 3\).
The term 'exact solutions' means you give the most accurate answer possible without approximations. Here are some key points to keep in mind:

• Solutions are considered 'exact' because they are not rounded or estimated.
• Use of math fractions, radicals, and symbols ensure precision.

For \(9y^2 = 49\), the exact solutions are \(y = \frac{7}{3}\) and \(y = -\frac{7}{3}\). These represent the points where the quadratic equation equals zero, ensuring the problem is solved and checked with absolute precision.