Problem 20 - Elementary Algebra

Question

For the following problems, solve each conditional equation. If the equation is not conditional, identify it as an identity or a contradiction. $$ 2 y+7=-3 $$

Step-by-Step Solution

Answer

Answer: \(y=-5\)

Step 1: Identify the terms on both sides of the equation

The equation is \(2y+7=-3\). Here, we have \(2y+7\) on the left side of the equation and \(-3\) on the right side.

Step 2: Isolate the y term

In order to isolate the y term, we will first move the constant term from the left side to the right side of the equation by subtracting 7 from both sides. This will give us: $$ 2y + 7 - 7 = -3 - 7 $$

Step 3: Simplify both sides

Now, we have: $$ 2y = -10 $$

Step 4: Solve for y

To find the value of y, we will divide both sides of the equation by 2: $$ \frac{2y}{2} = \frac{-10}{2} $$

Step 5: Simplify to get the value of y

The result is: $$ y = -5 $$ So, the solution to the equation \(2y+7=-3\) is \(y=-5\).

Key Concepts

Equation Solving

When we talk about solving equations, we're simply finding values that satisfy the given mathematical statement. In the exercise provided, the equation is \(2y + 7 = -3\). Our goal here was to find the value of \(y\) that makes this equation true.

• The first step is to rearrange the equation to isolate \(y\), by moving constants to one side.
• Then, simplify and further solve for \(y\).

Equation solving involves logical steps that simplify complex expressions and help us understand what is happening between the variables. The simplicity in breaking down each step, like moving terms across the equation and simplifying, provides clarity.

Identity

An identity is an equation that holds true for any value substituted into the variable. However, the equation \(2y + 7 = -3\) is not an identity.

• This equation has only one specific solution, \(y = -5\), making it a conditional equation.
• Identities, on the other hand, look like \(x + 2 = x + 2\), true no matter what \(x\) is.

Identifying if an equation is an identity helps us know if there’s a single solution or infinite solutions.

Contradiction

A contradiction arises in equations when no possible solution exists for the variable to satisfy the equation. For example, an equation like \(x + 3 = x + 5\) would imply that \(3 = 5\), which is not possible.

• Such equations signify that no solution exists.
• The initial problem didn't result in a contradiction since a valid solution \(y = -5\) was found.

Understanding contradictions highlights the importance of checking solutions in context, ensuring the logical consistency of the equation.

Isolating Variables

Isolating variables is the key step in solving equations. It means getting the variable by itself on one side of the equation. Let's see how it works:

• Start by moving any terms that aren't the variable to the opposite side, using arithmetic operations. In this case, subtracting 7 from each side simplifies \(2y + 7 = -3\) to \(2y = -10\).
• Once simplified, solve by performing inverse operations, like dividing both sides by 2 to isolate \(y\).

The process of isolating variables helps maintain the balance of the equation while clearly identifying the unknown value.