Problem 62 - Introductory Algebra for College Students

Question

Solve $$-14 y=-168$$

Step-by-Step Solution

Answer

The solution to the equation is \(y = 12\)

Step 1: Identify the Equation

The equation given is: \(-14y = -168\). This is a linear equation in one variable.

Step 2: Isolate y

To solve for y, we need to isolate y. We can do that by dividing both sides of the equation by -14. Remember, in mathematics, performing the same operation on both sides of an equation doesn't affect its equality. Hence, our new equation becomes: \(y = \frac{-168}{-14}\)

Step 3: Simplify the Equation

Simplifying the equation gives us: \(y = 12\)

Key Concepts

Solving Equations

Breaking down a problem into steps can make it easier to solve equations. In the world of mathematics, an equation is a statement that asserts the equality of two expressions. To solve it means to find the value of the unknown that makes the equation true. It requires isolating the variable you are solving for on one side of the equation.
Understanding the balance concept is crucial. Just like a balanced scale, both sides of the equation need to remain equal as you manipulate it. This balance can be achieved through:

• Addition or subtraction
• Multiplication or division
• Combination of these operations

Every step in solving the equation must maintain this balance until you find the solution for the unknown variable.

One Variable Equations

One variable equations, like \(-14y = -168\), involve only one unknown quantity, which simplifies things a bit. The goal here is to find the value of the variable that ensures both sides of the equation are equal.
Starting with the equation \(-14y = -168\), notice that the variable \(y\) is multiplied by \(-14\). Thus, our aim is to 'unwind' this process, getting \(y\) to stand alone.
Upon isolating the variable, what's left is a straightforward equation where \(y\) equals a specific number. This makes learning one variable equations a fantastic stepping stone before tackling more complex equations with multiple variables.

Division in Algebra

Division is a common operation used to isolate the variable in algebra. In the example -14y = -168, we need to use division to solve for \(y\). To remove the \(-14\) attached to \(y\), divide both sides by \(-14\).
This step results in \(y = \frac{-168}{-14}\), simplifying to \(y = 12\). A key point to remember is that when both sides are negative, dividing them results in a positive answer.
To ensure success with division in algebra:

• Always perform the same operation on both sides.
• Double check for signs (positive/negative) as it affects the final answer.
• Recheck your division to confirm the solution satisfies the original equation.

Develop comfort with division in algebra to build a strong foundation for solving various types of equations.