Problem 35 - Algebra 1: Concepts and Skills

Question

Solve the equation. $$ y-12=12 $$

Step-by-Step Solution

Answer

The solution to the equation is \(y = 24\).

Step 1: Identify the equation

The given equation is \(y - 12 = 12\). The goal is to isolate \(y\) on one side.

Step 2: Add 12 to both sides

The equation is re-arranged by adding 12 to each side to isolate \(y\). After doing so, the equation becomes \(y - 12 + 12 = 12 + 12\).

Step 3: Simplify Both Sides

Now simplify both sides of the equation to get the value of \(y\). On the left side, -12 + 12 equals 0 and leaves us only with \(y\). On the right side, 12 + 12 equals 24. The final equation is then \(y = 24\).

Key Concepts

Isolating Variables

When we talk about isolating variables, we mean getting one specific variable alone on one side of an equation. This process is essential because it helps us solve the equation and find the value of the variable. In the equation \(y - 12 = 12\), our goal is to isolate \(y\). To do this, we remove any numbers or expressions affecting \(y\) by using algebraic operations.

By isolating \(y\), you essentially "undo" what has been done to \(y\). Here, \(-12\) is subtracted from \(y\), so the opposite operation is adding \(12\) back. This cancels out the \(-12\) and leaves \(y\) by itself. Understanding how to isolate variables is a fundamental skill in solving equations.

Addition Property of Equality

The addition property of equality is a basic principle in algebra. It states that if you add the same number to both sides of an equation, the equality is still true. This property helps keep equations balanced.

In the equation \(y - 12 = 12\), we apply this property by adding \(12\) to both sides. So, it becomes \(y - 12 + 12 = 12 + 12\). Adding \(12\) to both sides ensures that \(y\) is isolated without altering the balance of the equation.

• This property is crucial because it allows us to manipulate an equation without changing its meaning or truth.
• It ensures that whatever operation you perform on one side, you must equally perform on the other side to maintain equilibrium.

The addition property of equality is one of the many tools that help simplify complex equations.

Linear Equations

Linear equations are a type of equation where the highest power of the variable is one. They are straightforward to solve and are often written in the form \(ax + b = c\).

The equation \(y - 12 = 12\) is a simple linear equation with one variable, \(y\). Solving linear equations often involves using basic operations such as addition, subtraction, multiplication, or division to isolate the variable.

• Linear equations can have one solution, no solution, or infinitely many solutions.
• They represent straight lines when graphed on a coordinate plane.

These equations form the foundation of algebra and are critical for understanding more complex mathematical concepts. Knowing how to manipulate and solve linear equations is an important skill that aids in problem-solving across different math areas.