Problem 56 - Elementary Algebra

Question

Solve. $$ 6 y-13=3+7 y $$

Step-by-Step Solution

Answer

The solution is \(y = -16\).

Step 1: Set up the Equation

Start with the given equation: \( 6y - 13 = 3 + 7y \).

Step 2: Simplify Both Sides

We'll first collect like terms or isolate variables on one side. To do that, subtract \(6y\) from both sides: \[ -13 = 3 + y \]

Step 3: Isolate the Variable

Now, subtract 3 from both sides to solve for \(y\): \[ -16 = y \]

Step 4: Verify the Solution

To ensure the solution is correct, substitute \(y = -16\) back into the original equation: - Replace \(y\) with \(-16\) in \(6y - 13 = 3 + 7y\). - Compute each side separately: - Left side: \(6(-16) - 13 = -96 - 13 = -109\) - Right side: \(3 + 7(-16) = 3 - 112 = -109\). - Since both sides equal \(-109\), the solution is verified.

Key Concepts

Solving Equations

Solving equations involves finding the value of a variable that makes the equation true. In the exercise given, we are working with a linear equation. A linear equation is an equation that involves only linear terms, which means the variables are raised only to the first power.
To solve such equations, you should aim to isolate the variable on one side of the equation. One common method is to use the technique of balancing equations.
This method includes:

• Performing the same operation on both sides of the equation. For instance, if you subtract a term from one side, you must subtract it from the other side too.
• Collecting like terms together, which means gathering all variables on one side and constants on the other.

In our problem, by subtracting like terms on both sides, we finally isolate the variable to get our answer. The key takeaway is to apply operations equally on both sides to maintain balance.

Verification of Solutions

Verification is a crucial step to ensure that the solution we find is correct. This involves plugging the solution back into the original equation to see if the equality holds.
Here's how the verification process works:

• Take the solution you found. In this exercise, it was determined that \(y = -16\).
• Substitute this value back into every occurrence of the variable in the original equation.

After substituting, you compute the resulting expressions on both sides of the equation. If both expressions evaluate to the same number, the solution is verified as correct.
In this case, both sides equaled \(-109\), confirming that \(y = -16\) is indeed the correct solution. This process might seem tedious, but it's an essential step to ensure accuracy.

Simplifying Expressions

Simplifying expressions is all about making the math easier to work with by combining like terms or performing arithmetic operations. In the context of solving equations, simplification helps prepare an equation for finding the solution.
In our problem, simplification involved several steps:

• First, we subtracted \(6y\) from both sides to gather all \(y\) terms on one side.
• We then performed arithmetic operations, like subtracting numbers, to further simplify.

This simplification leads to a much clearer picture of the variable's value.
Always remember, the more you simplify the clearer you make the equation, thus taking you a step closer to the solution. Simplification reduces mistakes and enhances clarity by stripping away unnecessary terms.