Problem 15 - Algebra A Combined Function

Question

Solve each formula for the specified variable. \(3 x+y=7\) for \(y\)

Step-by-Step Solution

Answer

The solution is \(y = 7 - 3x\).

Step 1: Understand the Equation

We have the linear equation \(3x + y = 7\). Our task is to solve this equation for the variable \(y\). This means we need to express \(y\) in terms of other variables or constants in the equation.

Step 2: Isolate the Variable y

To solve for \(y\), we need to isolate it on one side of the equation. We do this by subtracting \(3x\) from both sides of the equation to move terms involving \(x\) to the right side.

Step 3: Perform the Subtraction

Subtract \(3x\) from both sides of the equation:\[3x + y - 3x = 7 - 3x\]This simplifies to:\[y = 7 - 3x\]

Step 4: Rewrite the Solution

Now we have the solution in the form of \(y\). The equation \(y = 7 - 3x\) shows \(y\) expressed in terms of \(x\).

Key Concepts

Isolating Variables

In mathematics, especially in algebra, isolating a variable is a crucial skill. It involves rearranging an equation so that a particular variable stands alone on one side of the equation. When solving for a specific variable, like in our example where we need to solve for \(y\), the goal is straightforward: get \(y\) by itself on one side. This often requires moving other terms to the opposite side of the equation.
To isolate \(y\) in the equation \(3x + y = 7\), we perform inverse operations that undo addition or subtraction. Subtracting \(3x\) from both sides effectively removes the \(3x\) from the left side, leaving \(y\) isolated. Now, \(y\) is expressed solely in terms of \(x\) and constants. The process emphasizes symmetry and balance, ensuring that what you do to one side of the equation, you do to the other. This keeps the equation valid.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying equations or expressions, making them easier to work with. In the context of solving linear equations, it typically involves operations like addition, subtraction, multiplication, or division to achieve a desired form.
In our example, we're solving \(3x + y = 7\) for \(y\). The key is to "manipulate" the equation so that \(y\) is isolated. This involves looking at the equation and determining what operation will simplify or rearrange it to meet your goal. Here, subtraction is used to cancel out the \(3x\) term.
Algebraic manipulation is not just about performing operations; it's about understanding the relationships between terms. This understanding helps in knowing which operation to apply and in what order. Simplifying expressions and rearranging them systematically allows us to solve equations efficiently.

Equation Solving Process

Solving equations is a fundamental skill in algebra that involves using a series of logical steps to find the value of unknown variables. The process is a systematic approach to ensure every valid solution is found and verified.
We started with the linear equation \(3x + y = 7\), intending to solve for \(y\). The steps included:

• Clearly identifying the variable to solve for, in this case, \(y\).
• Applying algebraic techniques to isolate \(y\) on one side.
• Using inverse operations to maintain balance in the equation.
• Simplifying the expression to reach the solution \(y = 7 - 3x\).

The equation solving process we used illustrates how starting with a clear goal (isolating \(y\)) and knowing the appropriate operations lead to solving the equation successfully. This organized method can be applied to various equations to find solutions accurately and effectively.