Problem 42 - Elementary Algebra

Question

Solve. $$ 12 y+1=1 $$

Step-by-Step Solution

Answer

The solution is \( y = 0 \).

Step 1: Subtract 1 from both sides

We start by isolating the term with the variable on one side of the equation. Subtract 1 from both sides to maintain equality.\[ 12y + 1 - 1 = 1 - 1 \] This simplifies to:\[ 12y = 0 \]

Step 2: Divide both sides by 12

Next, we need to solve for \( y \) by dividing both sides of the equation by 12 to get rid of the coefficient of \( y \).\[ \frac{12y}{12} = \frac{0}{12} \]This simplifies the equation to:\[ y = 0 \]

Key Concepts

Equation Solving

Equation solving is a central part of algebra, and it involves finding the value of the unknown variable that makes the equation true. When solving equations, we aim to balance both sides of the equation by applying inverse operations. Inverse operations, such as addition and subtraction or multiplication and division, help to isolate the variable. This ensures that the operations performed on one side are mirrored on the other to maintain equality.

• Start by identifying the equation and understand what is being asked.
• Look for ways to simplify the equation if possible.
• Apply opposite operations to isolate the variable and solve the equation.

Equation solving often requires a series of steps and a logical progression to narrow down the variable's value. Keep practicing different techniques like grouping similar terms and using the distributive property to enhance your skills.

Linear Equations

Linear equations are equations of the first degree, meaning they have the highest exponent of the variable as 1. They are represented in the form \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable. Linear equations have a simple solution structure, as they form a straight line when graphed.

• The main feature of linear equations is that they produce one solution.
• Linear equations do not have squared variables or any variable raised to a power greater than one.
• These equations are often solvable in a few steps, focusing on isolating the variable.

Understanding linear equations lays the groundwork for exploring more complex mathematical concepts. Your ability to solve them depends on recognizing the components and accurately applying algebraic techniques. This understanding is crucial as more intricate algebraic expressions often rely on these foundational concepts.

Isolation of Variables

Isolation of variables is a strategy used in algebra to find the value of a variable in an equation by itself on one side. This is key to solving equations because once the variable is isolated, you can determine its exact value.

• You begin by performing inverse operations to move other terms away from the variable.
• For addition or subtraction, you use subtraction or addition, respectively.
• For multiplication or division, you will perform division or multiplication.

In the equation \(12y + 1 = 1\), we start by subtracting 1 from both sides, which eliminates the constant term, leaving \(12y = 0\). Next, dividing both sides by 12 isolates \(y\), giving us \(y = 0\). This process illustrates the importance of patience and precision. Isolation helps to simplify the equation, making it easier to visualize and solve.