Problem 70 - Algebra 2

Question

Solve each equation for \(y\). $$ 4 x+y=2 $$

Step-by-Step Solution

Answer

Solve for \(y\): \(y = 2 - 4x\).

Step 1: Identify the Goal

To solve the equation for \(y\), we want to rewrite the equation so that \(y\) is isolated on one side of the equation.

Step 2: Rearrange the Equation

Start with the given equation: \(4x + y = 2\). To isolate \(y\), we need to move \(4x\) to the other side. To do this, subtract \(4x\) from both sides of the equation.

Step 3: Simplify the Equation

After subtracting \(4x\) from both sides, the equation becomes: \(y = 2 - 4x\). This places \(y\) alone on the left side, and the expression with \(x\) on the right side.

Key Concepts

Solving for a Variable

When faced with an equation like \(4x + y = 2\), the task is to solve it for a specific variable, in this case, \(y\). Solving for a variable means re-writing the equation so that only that variable is on one side, typically the left. This allows us to express that variable in terms of the others present in the equation.

Here's how it works:

• Identify the variable you need to isolate. This is your target variable—in this exercise, it's \(y\).
• Make sure that \(y\) is by itself on one side of the equation.
• Move all other terms to the opposite side by performing equivalent mathematical operations (like addition or subtraction).

This process not only finds the value or expression for \(y\) but also helps understand how properties of equations work, which is vital for more complex algebra problems.

Isolation of Terms

Isolating terms is a fundamental step in solving equations. It involves moving all parts of the equation except the target variable (\(y\) in our case) away from one side of the equation. The idea is to leave the variable alone and free of interference from other terms.

In the original equation \(4x + y = 2\), you want to have \(y\) by itself. To do that, you will:

• Identify which terms need to be moved. Here, it's \(4x\) that needs to be removed from the side with \(y\).
• Use mathematical operations that do not change the equality of the equation, like subtraction or division, to remove these other terms. Subtract \(4x\) from both sides in this example.

After the isolation process, the equation will clearly show \(y\) by itself, which leads you to the solution \(y = 2 - 4x\). This process is crucial, as it reduces the complexity of an equation, making it easier to solve.

Rearranging Equations

Rearranging equations is about changing the structure of an equation to make it simpler or to solve it for a particular variable. This process highlights the dynamic nature of algebra, showing how flexibility and order of operations can alter the appearance of an equation while preserving its equality.

With an original equation like \(4x + y = 2\), rearranging involves:

• Reordering terms through appropriate algebraic operations—like moving \(4x\) to the right side of the equation to better isolate \(y\).
• Maintaining balance in the equation by performing the same operation on both sides.

This rearrangement helps in understanding the relationship between the variables, providing insight into how altering one can affect the others. It's a technique that not only aids in solving the present equation but also builds a foundation for solving more complex systems of equations.