Problem 37 - Elementary Algebra

Question

Solve. $$ 49 y=-23 $$

Step-by-Step Solution

Answer

\( y = \frac{-23}{49} \)

Step 1: Identify what you're solving for

The equation provided is \(49y = -23\). We are tasked with solving for the variable \(y\). This means we need to isolate \(y\) on one side of the equation.

Step 2: Isolate the variable

To isolate \(y\), divide both sides of the equation by 49. This is because \(y\) is currently being multiplied by 49. Dividing both sides will cancel out the 49 on the left side.\[ y = \frac{-23}{49} \]

Step 3: Simplify the fraction

Check if the fraction can be simplified. The numbers -23 and 49 have no common factors other than 1, so the fraction \( \frac{-23}{49} \) is already in its simplest form.

Key Concepts

Solving Linear Equations

When tackling the task of solving linear equations, your goal is to find the value of the unknown variable that makes the equation true. Linear equations are those in which the variable is only raised to the power of one, and no higher. Here's a simplified look at the process:

• Identify the unknown you need to solve for.
• Use operations such as addition, subtraction, multiplication, or division to isolate the variable on one side of the equation.
• Ensure that you perform the same operation on both sides to keep the equation balanced.

In the example given, "49u = -23," you are solving for "y." The key to solving any equation is maintaining balance, much like balancing a scale. Whatever you do to one side of the equation, you must do to the other.

Isolating Variables in Linear Equations

Isolating the variable means getting the variable by itself on one side of the equation, preferably the left side. This enables you to find the actual value of the variable. The steps to isolate the variable include:

• Identifying operations applied to the variable (e.g., addition, subtraction, multiplication, or division).
• Applying the inverse of these operations to both sides of the equation to remove them from the variable.
• Making sure that the coefficient of the variable becomes 1 (if necessary).

For instance, in "49y = -23," the variable "y" is multiplied by 49. To isolate "y," divide both sides by 49, leading to "y = \( \frac{-23}{49} \)." This step is essential as it clears any multiplier from the variable, effectively "freeing" the unknown and making it stand alone.

Simplifying Fractions

Once the variable is isolated, you might end up with a fraction. Simplifying fractions is the process of reducing the numerator and the denominator to their smallest possible integers. This is done to make the number easier to work with and understand. Here are the steps to simplify:

• Check if the numerator and denominator have any common factors other than 1.
• Divide both by any common factors.
• If no common factors are found, the fraction is already simplified.

In "y = \( \frac{-23}{49} \)," look for common factors between -23 and 49. Since they have no common factors other than 1, this fraction is already in its simplest form. Simplifying when possible helps prevent calculation errors in future steps and keeps solutions neat.