Problem 88 - Algebra 1

Question

Solve for \(y\). $$6 x-3 y=21$$

Step-by-Step Solution

Answer

The solution to the equation for \(y\) is \(y = 2x - 7\).

Step 1: Distribute factors

The given equation is \(6x - 3y = 21\). In this step, divide every term in the equation by 3 to simplify it. This yields \(2x - y = 7\).

Step 2: Isolate the term with \(y\)

Now isolate \(y\) on one side of the equation. To do this, subtract \(2x\) from both sides of the equation to obtain \(-y = -2x + 7\).

Step 3: Solve for \(y\)

You are almost done, there is just one more step to do. Multiply both sides of the equation by -1 to get \(y\) alone on the left side. This gives the final answer as \(y = 2x - 7\).

Key Concepts

Algebraic Manipulation

Algebraic manipulation involves rearranging and transforming algebraic expressions to simplify the problem or make it more manageable. When dealing with linear equations like the one given in the exercise, we often need to adjust terms to achieve our goal, which, in this case, is solving for a specific variable.
One basic type of algebraic manipulation is moving terms from one side of the equation to the other. This can involve addition, subtraction, multiplication, or division to isolate terms.

• To simplify the equation, divide all terms by the greatest common factor. In this problem, dividing every term of the equation \(6x - 3y = 21\) by 3 is a clear first step.
• By doing so, we managed to transform the equation into \(2x - y = 7\), which is more straightforward.

Algebraic manipulation is crucial in breaking down complex equations into forms that are easier to solve, helping us see the path to the solution more clearly.

Solving for a Variable

Solving for a variable means isolating the variable on one side of the equation, showing it directly in terms of the other variable(s). This concept is fundamental in algebra, as it allows us to understand the relationship between the variables in an equation.
In our exercise, we need to solve for \(y\) in the equation \(2x - y = 7\). To achieve this, we must isolate \(y\) to express it clearly in terms of \(x\).

• The first step is to move the non-\(y\) term \(2x\) to the other side of the equation, which is done by subtracting \(2x\) from both sides. We end up with \(-y = -2x + 7\).
• Obtaining \(y\) alone requires us to modify the equation further by multiplying both sides through by -1. This operation switches all the signs, resulting in \(y = 2x - 7\).
• This final form tells us how \(y\) depends on \(x\), providing the solution in a straightforward manner.

Understanding how to effectively isolate a variable empowers you to tackle a wide range of algebraic problems with confidence.

Equation Simplification

Equation simplification is the process of making an equation easier to understand and solve. This often involves reducing the complexity of the equation by removing excess or unnecessary terms and combinations. Simplification can make the next steps towards the solution much more obvious.
In our exercise, simplification began with the reduction of the equation by dividing each term by the common factor of 3, going from \(6x - 3y = 21\) to \(2x - y = 7\).

• By eliminating the factor of 3, the equation becomes less cluttered, simplifying further operations necessary to solve for a variable.
• The simplification helps us see that only essential steps remain – isolating \(y\) and expressing it in terms of \(x\).
• The entire process ultimately leads to a neater solution: \(y = 2x - 7\).

Effective equation simplification not only aids in immediate problem-solving but also enhances comprehension, making complex mathematical concepts more accessible.